Sir Alex Ferguson's Premier League table-toppers went into Saturday's late game buoyed by Manchester City's defeat to Everton in the early clash.
And they took the lead on 21 minutes, when Wayne Rooney opened the scoring with a deflected shot from 18 yards.
Managerless Reading showed great resolve, though, and kept United at bay for the remaining 70 minutes.
But they are still in deep relegation trouble, lying seven points from safety after Villa's massive win at Villa Park.
Villa, who beat Reading last time out, fell behind but eventually ran out 3-2 winners to open up a six-point gap on the bottom three, while Rangers remain rooted to the foot of the table.
Jermaine Jenas struck his second goal in as many games to give Harry Redknapp's side the lead but Gabriel Agbonlahor's header levelled matters on the stroke of half-time.
The hosts had been on the back foot for much of the game but found themselves in front when Andreas Weimann struck on the hour.
QPR showed their new-found belief under Redknapp and equalised through Andros Townsend before Christian Benteke scored a vital winner for Paul Lambert's side - his 17th goal of the season.
Reigning champions Manchester City failed to close the 12-point gap on leaders Manchester United after losing 2-0 at 10-man Everton in Saturday's early kick-off.
Leon Osman and Nikica Jelavic struck in either half for the Toffees, who had Steven Pienaar dismissed after picking up two yellow cards.
There was controversy late on as City should have been awarded a penalty for a Marouane Fellaini handball with the scoreline at 1-0.
But, although referee Lee Probert spotted Fellaini's misdemeanour, he only gave a free-kick with replays showing the Belgium international was well inside the area when he blocked Carlos Tevez's strike.
Southampton recorded an impressive 3-1 win over Liverpool at St Mary's in a game they dominated for large periods.
The hosts opened the scoring early on through Morgan Schneiderlin and Liverpool-born Rickie Lambert should have doubled Southampton's advantage but shot straight at Brad Jones before the striker saw a deflected free-kick make it 2-0 12 minutes before the break.
Philippe Coutinho cut Liverpool's deficit before the break but Jay Rodriguez scored a tremendous individual effort to wrap up the points and Mauricio Pochettino's second win since replacing Nigel Adkins as Saints' manager.
Arsenal closed to within two points of the Champions League places with a 2-0 win at Capital One Cup winners Swansea.
A first Arsenal goal for left-back Nacho Monreal and a strike from substitute Gervinho secured the three points after Alex Oxlade-Chamberlain rattled the crossbar twice for Arsene Wenger's side in the first half.
Stoke and West Brom shared a goalless draw at the Britannia Stadium.
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