Chelsea, with their inferior goal difference, knew a win was their only realistic chance of keeping alive their hopes of overhauling United - and they started the match powerfully.
Although the Blues were much the better side in the first half, Cole's volley against the bar aside, they created few clear-cut chances.
Whether United were affected by the loss of Vidic, who was stretchered off on 11 minutes after catching a knee in the face from Drogba, is debatable.
United's reshuffled defence, which saw substitute Owen Hargreaves come on at right-back and Wes Brown moved to centre-back, had coped well with Chelsea's threats.
But questions will be asked as to how the German international was allowed to ghost in unmarked at the far post to score the opener.
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Grant says Chelsea dominated the game
Michael Essien's running from deep was causing United problems, but too often the midfielder was not played in when in a good position as Chelsea made the wrong decisions.
But the power of Drogba, the combative qualities of Essien and Cole's darting movement had United pinned back in their own half.
And when United did find an outlet, often in right winger Nani, they failed to make any impact as time and again Chelsea snuffed out a limp attack and hit back themselves.
A quiet Rooney looked isolated, and United seemed toothless without Ronaldo, who had been left on the bench.
After the interval, Chelsea again tried to pick up the pace but their game plan appeared to be in tatters when Carvalho made an awful mistake to let United back into the match.
The Portuguese defender casually played the ball towards his own goal, straight into the path of Rooney, who raced clear of John Terry and scored with an excellent low drive.
It proved to be Rooney's last contribution as he was went off injured shortly after to be replaced by Ronaldo.
Regardless, United were now in the ascendancy and Chelsea looked shell-shocked.
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Man Utd assistant Carlos Queiroz felt both penalty decisons were wrong
Ryan Giggs had a fierce shot well saved by Petr Cech and United were stroking the ball around confidently.
Frustration was building for Chelsea and Ballack and Drogba argued with each other as they prepared to take a free-kick.
Drogba eventually took aim and was only denied by a magnificent save by Edwin van der Sar.
And there was still time for more drama as Carrick was penalised for blocking Essien's right-wing cross with his arm.
Up stepped Ballack, who brilliantly sent the keeper the wrong way and put Chelsea back into the lead.
Yet it almost did not end there as United came so close to snatching another equaliser but Cole and Shevchenko cleared efforts off the line.
Chelsea:
Cech, Ferreira (Anelka 66), Carvalho, Terry, Ashley Cole, Ballack, Obi, Essien, Kalou (Shevchenko 81), Drogba, Joe Cole (Makelele 87).
Subs Not Used: Cudicini, Belletti.
Booked:
Ballack, Obi, Drogba.
Goals:
Ballack 45, 86 pen.
Man Utd:
Van der Sar, Brown, Ferdinand, Vidic (Hargreaves 14), Silvestre, Fletcher, Carrick, Anderson (O'Shea 65), Nani, Rooney (Ronaldo 63), Giggs.
Subs Not Used: Kuszczak, Tevez.
Booked:
Brown, Van der Sar, Ferdinand, Hargreaves.
Goals:
Rooney 57.
Att:
41,828
Ref:
Alan Wiley (Staffordshire).
BBC Sport Player Rater man of the match:
Chelsea's Michael Ballack 7.85 (on 90 minutes).
Please note that you can still give the players marks out of 10 on BBC Sport's Player Rater after the match has finished.
