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Sir Alex Ferguson's team looked set to suffer their first league defeat since April when Ashley Young's second half-penalty was followed by a Marc Albrighton strike for Villa.
But United substitute Federico Macheda reduced the deficit with a superb strike nine minutes from full-time and Serbia defender Nemanja Vidic dived in to head the equaliser from Nani's cross four minutes later.
Although United were understandably delighted to escape Villa Park unscathed, Ferguson's second-placed side, who are now three points behind Chelsea, will still be left to count the cost of a sloppy display if the leaders open up a six-point gap by beating Sunderland on Sunday.
"In the second half I just think it was carelessness on our part. Our passing was poor. Normally we can control the match. We didn't control it," Ferguson said.
"We are a club, though, that never says there is no way back. We brought people on and they changed the game in fairness to them.
"Macheda scored a great goal and with another five minutes we would have won the game."
Manchester City fans turned on Roberto Mancini after the Italian's side were held to a dour 0-0 draw by Birmingham at Eastlands.
City boss Mancini, whose team have failed to score in their last three home games, provoked a furious response when he decided to replace Argentine forward Carlos Tevez with midfielder Gareth Barry as the hosts tried in vain to break down a stubborn Birmingham defence.
Tevez earlier had a goal disallowed for handball and fourth-placed City were unable to close the gap on Chelsea.
Gareth Bale was in fine form as Tottenham beat Blackburn 4-2 at White Hart Lane.
Wales winger Bale was linked with some of Europe's biggest clubs after his recent performances against Inter Milan in the Champions League and he underlined his growing reputation with another dominant display.
He opened the scoring in the 16th minute when he headed home Rafael van der Vaart's corner.
Harry Redknapp's side squandered a golden chance to increase their lead as Roman Pavlyuchenko smashed a penalty wide after Rovers goalkeeper Paul Robinson fouled Peter Crouch.
But Russian striker Pavlyuchenko made amends for that howler in the 42nd minute as he headed Bale's cross past Robinson.
Crouch netted Tottenham's third with a 69th minute tap-in before Bale struck again six minutes later.
Ryan Nelsen netted for Blackburn in the 80th minute and Gael Givet added a second in the 90th minute.
Bolton moved into fifth place in the table with a 3-2 win at struggling Wolves.
Owen Coyle's side were ahead after just 45 seconds as Richard Stearman, under pressure from Matt Taylor, could only turn the ball into his own net.
Johan Elmander increased their lead in the 62nd minute with a superb solo effort that saw him dribble past four defenders before slotting home.
Stuart Holden put Bolton in total control with his first goal for the club in the 67th minute.
Kevin Foley's reply for second-bottom Wolves moments later was followed by a 77th-minute strike from Steven Fletcher but Mick McCarthy's team couldn't complete the comeback.
Victor Moses fired Wigan out of the relegation zone as his 70th-minute strike secured a 1-0 win over West Bromwich Albion at the DW Stadium.
West Ham remain bottom of the table after Blackpool's first-teamers returned for a 0-0 draw at Upton Park.
Seasiders boss Ian Holloway made 11 changes to restore his stars to the line-up after controversially sending out a shadow side for the mid-week defeat at Aston Villa.
And the changes were enough to frustrate West Ham and keep Blackpool away from the relegation zone.
Newcastle dropped to seventh after a 0-0 draw at home to Fulham.