Assistant manager Simon Donnelly said Dundee United now have their sights firmly set on catching Inverness CT as they chase a Europa League spot.
The Terrors ended a run of 10 games without a win as Chris Erskine's second-half strike was enough to earn a 1-0 victory over Hamilton.
With Inverness drawing 1-1 with Celtic, the Tannadice side move to within six points of John Hughes' men with a game in hand, against Aberdeen.
"We've stated all along we want to challenge at the top," Donnelly told BBC Scotland.
"We've fell away a wee bit recently, but we've got third place in our sights and it's up to us to go and achieve that now.
"We've got Inverness to play twice and we've got a game in hand so it's in our own hands."
The home side had goalkeeper Radoslaw Cierzniak to thank for a string of saves with Hamilton the more threatening of the sides, especially in the second half.
And Donnelly admitted his players were relieved to finally get the three points that had eluded them recently.
He said: "I think I'd be lying if I said there wasn't [relief]. The run we've been on has been really difficult for everyone, and it's a much needed three points.
"It alleviates a bit of the pressure, and the boys are going away with a wee spring in their step tonight and we can look forward to the rest of the games.
"The boys to a man really fought for each other. We've shown a real desire today and a mental strength on a difficult pitch against a hard Hamilton team."
Hamilton player-manager Martin Canning told the Hamilton supporters their club are still in a good place despite going 13 games without a win, and said he has already started planning for next season.
"It's a dejected changing room in there," he said. "Today sums up the last 13 games. We're doing a lot of things right, the boys are following the plan, and we're creating opportunities but the ball just won't go in the net.
"They put so much in and they deserved to get something out of that game. It's so difficult to take, I think that's their first shot on goal in the second half, and maybe their only shot on goal in the second half.
"The quality of play has been good, we're just not getting the result. There's a lot of terrific young players in that changing room and they're gutted and rightly so because for their efforts this season they probably deserve to be in the top six.
"In terms of this club's future, the club's in a good place. In terms of the season as a whole it's been a great season for Hamilton.
"We've a lot of good young players, we're in the Premier League where we want to be, and we go again next season. We'll look to bring in the three or four boys that we need to make us a bit stronger again.
"We know what we need to do in summer to make us a potent threat again next year."