A first win for Arsenal in four games took them level on 48 points with second-placed Tottenham Hotspur, their bitter local rivals who have a superior goal difference.
Surprise package Leicester City are five points clear at the top after their 3-1 win at Manchester City on Saturday.
After more than five hours without a league goal, Arsenal scored two in little over a minute midway through the first half from Mesut Ozil and Alex Oxlade-Chamberlain.
Arsene Wenger's team, aiming for their first Premier League crown since 2004, host Leicester next Sunday.
"It was a must-win game for us today," Wenger told the BBC. "Leicester have made a big impression in the last week after their results. They have suddenly become the favourite in the Premier League and it makes our game (next week) a big game."
United played the better football at Stamford Bridge and went ahead through Jesse Lingard's superb strike on the hour but Chelsea's Diego Costa equalised in the first minute of added time to leave United fifth on 41 points from 25 matches.
They are 12 adrift of Leicester and six points behind the fourth Champions League spot occupied by Manchester City.
RAMSEY INVOLVED
Arsenal registered their first league win since Jan. 2, having played out two 0-0 draws and lost 1-0 to Chelsea, with Welsh midfielder Aaron Ramsey heavily involved in both goals.
In the 23rd minute he crossed for Olivier Giroud to head down and give Ozil the opportunity to score.
Bournemouth had not recovered before Ramsey sent Oxlade-Chamberlain through for his first league goal since September 2014 -- and first away from home -- with an angled drive.
Bournemouth, playing their first season in the Premier league, remain 15th, five points above the relegation places.
The Chelsea-Manchester United fixture had been one of the highlights of the season for most of the past decade but with both teams having struggled for form this season, there was a subdued atmosphere in the first half.
The game finally came to life when Lingard put United ahead with a superb goal, turning and shooting after a cross from the left had been touched on by Wayne Rooney.
Chelsea left it late but finally equalised when Costa rounded Spain team mate David De Gea to score as they drew at home for the fourth straight league match, leaving them in 13th place on 30 points, 17 points from the Champions League places.
United coach Louis van Gaal could not hide his frustration at conceding such a late goal after a promising performance.
"We played very well until maybe the last quarter and it was very frustrating they scored so late. We played like we played today but we are not making the gap smaller (on the Champions League places)," he said.
(Reporting by Steve Tongue and Mike Collett, editing by Ken Ferris/Toby Davis)
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