An own goal by Jonny Evans, a scorching Juan Mata volley and David Luiz's header six minutes into the second half put Chelsea in complete control but Rooney twice held his nerve from the spot and substitute Javier Hernandez headed the equaliser seven minutes from time to earn the champions a point.
Manchester City lead the standings with 57 points, two ahead of United and eight clear of Tottenham Hotspur who travel to Liverpool on Monday. Chelsea stayed fourth on 43 points.
Earlier Newcastle United beat Aston Villa 2-1 to climb above Arsenal into fifth place, keeping the race for the title and European places tightly congested after 24 matches.
Chelsea and United have been major rivals for the big prizes over the last few seasons and the match lived up to expectations in a heart-stopping classic at Stamford Bridge.
"Of course it's two points lost but, after we were 3-0 down, all the Manchester City players at home watching the game won't enjoy the way we fought back," Rooney told Sky Sports. "They can see we've got spirit and we will be there with them until the end."
Chelsea, who had not lost at home in the league to United since 2002, took the lead against the run of play when Evans put through his own goal after 36 minutes following clever work by Daniel Sturridge and they doubled the lead with a contender for goal of the season from Mata 25 seconds into the second half.
Mata converted a cross from his Spanish compatriot Fernando Torres, who has now gone 18 matches without a goal, when he smashed in an unstoppable left-foot volley.
They went 3-0 up soon afterwards when Luiz's header went in off Rio Ferdinand, booed by the home fans every time he touched the ball.
Ferdinand's younger brother Anton, the Queens Park Rangers defender, was the target of the alleged racist insluts that have landed Chelsea skipper John Terry in court on racial charges, but some of the heat was taken out of the contest when Terry was declared unfit with a knee injury.
Chelsea were also without the injured Frank Lampard and suspended Ashley Cole, but coach Andre Villas-Boas handed a debut to defender Gary Cahill and gave a first start of the season to fit-again midfielder Michael Essien.
Villas-Boas, whose side have now drawn their last three league games, said United's second penalty was "very dubious."
"I don't think the referee was in the correct position to see it, I am not sure. I don't know if he was compensating for anything in the first half but it was the wrong decision, Welbeck's leg hit Ivanovic," Villas-Boas told reporters.
"The penalties gave them the impetus to get the draw."
But he failed to condemn the Chelsea fans for booing Ferdinand, saying it had no impact on the result and that home fans always boo opposition players.
"You don't expect opposition fans to praise the other team's players, you have to condemn it when there is discriminatory or aggressive behaviour. We kno
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