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The 17-year-old Italian, coming off the bench after 60 minutes in place of Nani, turned and curled in a superb strike three minutes into stoppage time to mark his debut.
The crucial win lifted the champions back to the summit with 68points from 30 games and kept them well on course for the title.
United, with a game in hand, have one-point edge over second-placed Liverpool who temporarily went top on Saturday after an equally dramatic 1-0 win at Fulham.
Chelsea, who won 2-0 at Newcastle United Saturday, are third on64 points.
Macheda's winner put an incredible end to a match of twists and turns in which short-handed United had appeared to be heading for a third successive league defeat after surrendering to Liverpool and Fulham.
Villa, looking for their first win in nine matches, led 2-1 until 10 minutes from time but Cristiano Ronaldo restored the parity with his second before Macheda provided the surprise.
Villa have to remain fifth by a point behind Everton, who slaughtered Wigan Athletic 4-0 at home.
Martin O'Neill's men started the game brightly but it was United that opened fire first with Ronaldo scoring from an indirect free-kick after a James Milner back pass after 14 minutes.
United, however, suffered from the absence of injured Rio Ferdinand and Dimitar Berbatov and suspended Paul Scholes, Nemanja Vidic and Wayne Rooney as the visitor surged back with two headers.
John Carew leveled for Villa on 30 minutes, completing a quick counter with a powerful header into into the bottom corner, and Gabriel Agbonlahor headed Villa in front 13 minutes into the second half only to see the result against them.