Diego Costa scored a late equaliser to preserve Chelsea's unbeaten record under interim boss Guus Hiddink and deny Manchester United a win.
United dominated early on, Thibaut Courtois producing a fine one-handed save to keep out Anthony Martial.
David de Gea saved well from Nemanja Matic's header before Jesse Lingard's fine turn and shot put United ahead.
Costa rescued a point in the 91st minute after rounding De Gea following Cesc Fabregas' pass.
United were moments away from moving to within four points of the top four when Costa scored his seventh goal in eight games.
Chelsea, who lost defender Kurt Zouma to a serious injury, remain 13th in the table.
This will feel like a defeat for United after a bright start and a wonderful goal to give them the lead in what was an entertaining draw between two teams struggling for form.
Apart from the late equaliser, Louis van Gaal's side produced a display full of positives after recent criticism about the team's style and reports linking former Chelsea boss Jose Mourinho with the United job.
They started well and offered more zip and energy than their cumbersome hosts.
Courtois' flying save to keep out Martial's curling attempt was excellent, while United forced 10 corners before the end of the first half.
The visitors looked set for a rare win over Chelsea when Lingard displayed terrific balance and agility to break the deadlock with a sublime spin and shot inside the area.
However, United backed off after scoring and invited pressure, Costa pouncing from close range after a terrific Fabregas pass.
Instead of closing the gap on fourth place to four points, they now find themselves six points adrift.
De Gea was at his magnificent best to keep out a thunderous attempt by Branislav Ivanovic and then Fabregas as Chelsea surged forward in the final quarter in search of an equaliser.
When Costa blazed over the bar after a free-kick it looked all up for the hosts.
Yet for the second league home game in a row, they rescued a point in stoppage time.
Hiddink's sixth draw in eight league games in charge leaves the defending champions seven points above the relegation zone with 13 games remaining.
This was Chelsea's first game back at Stamford Bridge since captain John Terry announced he was set to leave at the end of the season.
There was no mention of Terry leaving in the captain's programme notes and Blues fans will still be hoping he may yet extend his stay.
They chanted his name throughout as Terry produced an assured performance at the heart of the defence, while he was denied a penalty when his goal-bound shot at the end of the first half struck the arm of Daley Blind.
Terry's importance to the team was underlined by Zouma's nasty-looking injury, the France international landing awkwardly on his right knee after volleying the ball away.
Chelsea manager Guus Hiddink:
"I am happy with the reaction of the team. Manchester United did a good job in the first half and in the beginning of the second, they are not an easy team to beat.
"But we deserved a point. We dropped too far back after the first 20 minutes when they dominated us but later on we closed a little more.
"They made a beautiful goal - without good marking from our side - but after I think the team reacted very well."
Manchester United manager Louis van Gaal:
"I feel we didn't reward ourselves with a victory because we were the better team. Until the last quarter we played very well and could have scored more goals. But you have to control the game better and we did not do that.
"It isn't a lack of concentration it is not being composed when you are defending. When we have the ball we have to keep the ball.
"You need to give the right pass at the right moment and we did not do that [before Chelsea scored]. Chelsea are a very good team and you play like we played and don't reward yourself, it is frustrating."
Both teams are in Premier League action on Saturday at different times of the day. While Chelsea entertain struggling Newcastle United (17:30 GMT), Manchester United travel to Wearside to face relegation-threatened Sunderland (12:45 GMT).