Joleon Lescott’s second-half header, fumbled in by keeper Wayne Hennessey, handed Villa manager Remi Garde a first victory in charge, the bottom club’s first in the league at home this season, and ended a run of 19 games without a win.
Wayne Rooney’s 79th-minute strike, his second goal of the game, put United 3-2 up at Newcastle but Dummett struck in the last minute for the relegation-threatened hosts who had battled back from 2-0 down.
West Ham United moved above Manchester United into fifth place after coming from a goal down to win 3-1 at Bournemouth, Enner Valencia netting twice for the London side.
Villa, still eight points from safety, were gifted victory when Hennessey allowed defender Lescott’s 58th-minute effort to squirm through his hands.
The home team had appeared to be in for another 90 minutes of struggle when Palace’s Wilfried Zaha struck Villa’s post inside the opening minute but Garde’s men held firm.
The Villa players, abused by their fans after drawing with fourth-tier Wycombe Wanderers in the FA Cup at the weekend, this time received a standing ovation as they left the pitch.
“We suffered so much in the previous weeks and today we are very happy. Definitely the players showed pride and we deserved to win,” Garde told Sky Sports.
“It was a nice feeling to feel the crowd behind the players, it’s very important. We cannot be happy too long — we have to build on this one.”
Newcastle remain in the relegation zone, one point behind 17th-placed Swansea City, but manager Steve McClaren was delighted his side had shown “what we are about”.
McClaren’s side trailed 2-0 after Rooney converted a ninth-minute penalty and laid on the second for Jesse Lingard following a swift counter attack.
Aleksandar Mitrovic nodded down for Georginio Wijnaldum to pull a goal back just before halftime.
Serb Mitrovic then levelled from the spot after 67 minutes when Chris Smalling was penalised for pushing.
Rooney’s neat finish put United back in front but Dummett lashed the ball in from the edge of the box to earn his side a point.
“People said Manchester United were boring and we lacked character but we kept fighting until the end and we have got a goal that feels like a winner,” McClaren said.
“People thought it would be a 0-0 and it was a classic at 3-3.”
Dimitri Payet kickstarted West Ham’s comeback at Bournemouth after they had trailed to a fine first-half strike from Harry Arter.
West Ham lost striker Andy Carroll to injury early on but Payet levelled midway through the second half with a sublime free kick.
The French midfielder then set up Valencia to make it 2-1 in the 75th minute and the Ecuadorean striker thumped in a free kick to seal victory nine minutes later.
Leaders Arsenal visit Liverpool on Wednesday while second-placed Leicester City travel to Tottenham Hotspur.