Sunderland are one victory away from securing Premier League survival, while rivals Newcastle and Norwich are on the brink of relegation after slipping up.
Sam Allardyce's side climbed out of the bottom three by defeating Chelsea 3-2, and will be safe if they beat Everton on Wednesday.
"Our destiny is in our own hands. We have turned it around with a big victory," Allardyce said.
Newcastle drew 0-0 at Aston Villa and Norwich lost 1-0 to Manchester United.
Rafa Benitez's side, who have one game left, are a point behind Sunderland, while Norwich are four points adrift of safety and, like Sunderland, have two matches remaining.
'There were so many biting their nails'
Sunderland's spirit in coming from a goal down twice to beat Chelsea was crucial in giving them the advantage in the race for survival as the finish line approaches.
Allardyce's team have been difficult to beat since the turn of the year, losing just four of 17 league matches in 2016, but had struggled for wins, having drawn six of their eight games before Saturday.
They trailed Chelsea 2-1 at half-time, with Nemanja Matic scoring with Wahbi Khazri's brilliant volley equalising Diego Costa's opener.
Two quick second-half goals from Fabio Borini and Jermain Defoe turned the game in favour of Sunderland, before Chelsea captain John Terry's late sending off.
"Newcastle drawing and Norwich losing has put us in a very good position," Allardyce said. "Now we must take full advantage of that position and try to finish it off on Wednesday.
"I hope we will not jangle our nerves any longer than we need to. There were so many people biting their nails today in the last few minutes because it was such a big result."
|Sunderland - still to play|
|Weds, 11 May: Everton (h)|
|Sun, 15 May: Watford (a)|
'We were anxious'
Newcastle were poor as they failed to beat an Aston Villa side who had lost their previous 11 league matches.
It took Benitez's team 36 minutes to register a shot on target, although they did create more chances after half-time, with substitute Aleksandar Mitrovic lobbing over the bar when clean through, and heading wide with two other good opportunities.
"For us, it's bad, but simple," said Benitez. "We have to wait and then after try to win, depending on what Sunderland do during the week.
"If it goes to the last game, then we have to fight to get the three points and see what happens. Everyone is disappointed, but this is normal.
"We were a little bit anxious in the first half. We were not comfortable in possession. You have to take your chances and we didn't do that.
"We had to change things in the second half. We did that, but it was not enough. The players are disappointed."
|Newcastle - still to play|
|Sun, 15 May: Tottenham (h)|
'It's not over - but it's a blow'
Norwich looked well placed to stay up after beating Newcastle 3-2 on 2 April, but their survival chances have been all but extinguished by four successive defeats since then.
A 3-0 loss at home to Sunderland on 16 April was particularly damaging, with Alex Neil's side then losing at Arsenal before a 1-0 home defeat to Manchester United this weekend.
Juan Mata scored the only goal with 18 minutes to play, after Wayne Rooney had taken advantage of a Sebastien Bassong error to set up the chance.
"It's not over, but it's a major blow," Neil said. "It looks a real tough situation to get out of.
"There have been so many times across the season where we've been well in the game, played pretty well at certain stages.
"They've scored off the back of another mistake - and that's something that has been quite commonplace for us this year.
"It's tested my patience for a long time. They're not trying to do it, but when you're maybe under pressure and not performing, sometimes those errors can creep in."
|Norwich - still to play|
|Weds, 11 May: Watford (h)|
|Sun, 15 May: Everton (a)|
Who needs what?
Norwich will be relegated if they lose to Watford on Wednesday, or if they draw and Sunderland avoid defeat against Everton on the same night, or if Sunderland win.
Alex Neil's team need to better Sunderland's result to stay alive - but even a draw-plus-a-Sunderland-loss leaves them teetering as they would have to make up a swing in the region of nine goals on the final day.
Newcastle, who are not playing on Wednesday, will be relegated if Sunderland win.
Sunderland, by contrast, will be safe with a win, either against Everton in midweek or at Watford next weekend.
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