Match ends, Bournemouth 1, Manchester United 2.
Jose Mourinho said he was the "luckiest manager in the Premier League" in the first half as Manchester United needed a late Marcus Rashford goal to beat Bournemouth.
Rashford's dramatic injury-time goal snatched the three points for the Red Devils as he netted at the second attempt following Paul Pogba's cross.
United had been fortunate to be level at 1-1 at the interval after a lacklustre first-half display.
"The score should have been like 6-2," said Mourinho.
"We were defensively awful. As a team we didn't press or do any of the work we did in the week. People watching this game will not believe how hard we worked this week."
The Cherries had come close three times inside the first 10 minutes before a corner was half-cleared to Lewis Cook. The young England midfielder played a superb through pass to Junior Stanislas, whose pinpoint right-wing cross was stabbed in by Callum Wilson.
And the visitors were lucky not to be further behind by the time Ashley Young burst down the right and played in Alexis Sanchez, who cut it back for Anthony Martial to guide home.
Mourinho said "top teams don't start as bad" as United did and described the first-half performance as "chaotic".
"I'm upset with that. The second half was much better. We created lots of chances and deserved to score before the 92nd minute," he said.
United nearly took the lead after the break when Young rattled the frame of the goal with a free-kick. Rashford's goal-bound follow-up was cleared off the line by Nathan Ake, before David Brooks hacked Pogba's effort clear.
But Rashford eventually found a way through - and victory will lift Mourinho at the start of a week during which his side have two more away trips - to Juventus in the Champions League on Wednesday, and Manchester City next Sunday.
Man Utd labour on the road
Only a win by a four-goal margin would have allowed eighth-placed United to displace Bournemouth from their lofty position in the top six, but that did not look like happening as they were caught cold early on.
United goalkeeper David de Gea had already been called into action to block efforts by Ryan Fraser and Wilson before Cook opened up the defence and De Gea was given no chance by the in-form Wilson - who, like his manager, was nominated on Friday for a Premier League monthly awards.
While their defence was regularly threatened by the pace of Fraser and Wilson, United were disjointed in midfield, where there was a private battle between the two clubs' record signings to dominate the centre of the park, and attack.
Bournemouth's £25m capture Jefferson Lerma had the better of the £89m Pogba for much of the game, while Sanchez had a frustrating afternoon, often left isolated up front or beseeching referee Paul Tierney to give a decision in his favour.
But a double substitution 11 minutes after the interval improved United's fortunes, with Rashford and fellow replacement Ander Herrera both going close.
It was the home side who were hanging on by the end, although the visitors still threatened on the break and De Gea stood firm to keep out Brooks before Rashford's late winner.
Bournemouth still standing tall
It shows how far Bournemouth have come in their rise through the divisions that a one-goal defeat by Manchester United - in a league fixture whose very existence was unthinkable five years ago - must feel such like a kick in the teeth for their fans.
Indeed, based on the first quarter of the game, there was only one team here deserving an invitation to a European Super League, as United and others have reportedly been given - and it was not the visitors.
Despite having been sixth in the table after 10 games, it would be premature to suggest the Cherries are ready to pit their red and black striped shirts against those of AC Milan or other European opposition, but it would have been a different result if they had added a second goal during their period of first-half dominance.
Bournemouth - as ever, so much more than the sum of their parts under Eddie Howe - have the pace and skill in attack to threaten any Premier League team.
While they had to defend in numbers at times - with Steve Cook and particularly Nathan Ake standing tall in central defence until that late winner - Howe's men can count themselves unlucky not to have taken a point.
Between now and the end of December, Bournemouth face some stiff tests against the Premier League's elite - with six of their next 10 league and cup fixtures against Arsenal, Manchester City, Liverpool, Chelsea and Tottenham before they travel to Old Trafford for the return fixture with United on 30 December.
If they can maintain a top-half placing in the table by the time 2018 becomes 2019, it will be a considerable achievement for the Dorset club.
Man of the match - Marcus Rashford
'In the first half we were a lucky team'
Bournemouth boss Eddie Howe: "A tough one for us, a lot of our play was really good especially during that opening period. We created chances and we were disappointed we weren't winning at half-time as we deserved to lead by more than one.
"They made some good substitutions but I'm very pleased with the effort, That's one of the best first-half performances I've seen since we've been in the Premier League and I can't fault them in terms of effort.
"Full credit to Manchester United for sticking in the game, it couldn't have been easy for them, some of our movement was second to none. The first goal we conceded could probably have been avoided."
Manchester United boss Jose Mourinho: "When we make a defensive mistake, instead of forgetting it and keeping playing, it triggers instability - like after a couple of minutes of the game with Ryan Fraser facing David de Gea, that triggered instability.
"It looked like we don't work tactically during the week, which is the worst thing for me as we work hard. But we have the faith that the game is not over, in the second half I could understand the frustration as we created chances but couldn't score."
Last-gasp Rashford - the stats
- Bournemouth lost at home for the first time in seven Premier League games (W4 D2), since losing to Manchester United in April (0-2).
- No team has come from behind to win more points in the Premier League this season than United (seven, level with Arsenal and Leicester).
- United have kept just one clean sheet and conceded 18 goals after 11 Premier League games this season. At this stage last season they had registered eight clean sheets and shipped just five goals.
- Mourinho is unbeaten in his 21 Premier League meetings with English managers with United, winning 17 and drawing four.
- In 11 Premier League appearances this season, Wilson has already equalled his tally of goal involvements from last season (10), achieved in 28 games.
- Stanislas has had a hand in four goals in his past five Premier League home games (three goals, one assist), one more than he managed in his previous 14 (three).
- Rashford has scored two 90th-minute winning goals in Premier League games for Man Utd - no player has netted more for the club (level with Paul Scholes and Robin van Persie).
- 3S Cook
- 15A SmithBooked at 83mins
- 19StanislasBooked at 63minsSubstituted forSurmanat 90+1'minutes
- 16L CookSubstituted forGoslingat 68'minutes
- 8LermaBooked at 58mins
- 24FraserSubstituted forIbeat 74'minutes
- 13WilsonBooked at 64mins
- 1De Gea
- 18YoungBooked at 82mins
- 23ShawBooked at 54mins
- 17FredSubstituted forHerreraat 56'minutes
- 8MataSubstituted forRashfordat 56'minutesBooked at 90mins
- 7SánchezSubstituted forLingardat 78'minutes
- Paul Tierney
- Shots on Target
Second Half ends, Bournemouth 1, Manchester United 2.
Foul by Ander Herrera (Manchester United).
Simon Francis (Bournemouth) wins a free kick in the defensive half.
Marcus Rashford (Manchester United) is shown the yellow card for excessive celebration.
Goal! Bournemouth 1, Manchester United 2. Marcus Rashford (Manchester United) left footed shot from very close range to the centre of the goal.
Substitution, Bournemouth. Andrew Surman replaces Junior Stanislas.
Corner, Bournemouth. Conceded by Ashley Young.
Attempt saved. Jesse Lingard (Manchester United) right footed shot from the centre of the box is saved in the centre of the goal. Assisted by Marcus Rashford.
Attempt blocked. Paul Pogba (Manchester United) right footed shot from outside the box is blocked. Assisted by Jesse Lingard.
Offside, Manchester United. Marcus Rashford tries a through ball, but Jesse Lingard is caught offside.
Adam Smith (Bournemouth) is shown the yellow card.
Ashley Young (Manchester United) is shown the yellow card.
Attempt saved. David Brooks (Bournemouth) left footed shot from the left side of the box is saved in the bottom left corner. Assisted by Jordon Ibe.
Corner, Manchester United. Conceded by Steve Cook.
Substitution, Manchester United. Jesse Lingard replaces Alexis Sánchez.
Corner, Bournemouth. Conceded by Nemanja Matic.
Corner, Bournemouth. Conceded by Ander Herrera.
Substitution, Bournemouth. Jordon Ibe replaces Ryan Fraser.
Offside, Manchester United. Nemanja Matic tries a through ball, but Alexis Sánchez is caught offside.