Juan Mata had given United a deserved first-half lead against his former employers in what had been dubbed a "must-win" match for the hosts, but De Gea's mistake gifted an equaliser to Marcos Alonso as Chelsea boosted their hopes of Champions League qualification.
The result leaves Maurizio Sarri's side still fourth in the table and two points above fifth-placed Arsenal, whereas United must overhaul a three-point deficit as well as their inferior goal difference during the final two games of the season.
The importance of the game was increased by defeats for Tottenham Hotspur and Arsenal earlier in the weekend, and United in particular began like a team who knew their top-four hopes depended on winning.
The hosts created their first big chance after just four minutes when Luke Shaw's long pass forward beat the entire Chelsea defence to leave Romelu Lukaku through on goal, but the Belgian took too long over his finish and was eventually denied by Kepa Arrizabalaga.
United would not have to wait long to break the deadlock, though, as Mata marked his 31st birthday with a goal against his former club after a scooped Lukaku pass had released Shaw, who cut the ball back into the middle for Mata to convert.
Chelsea were largely limited to long-range efforts for much of the first half, with Alonso and Jorginho failing to test David de Gea when trying their luck.
De Gea would go on to fail the first test which came his way from another long-range strike, but United almost doubled their advantage before that when Eric Bailly came close to marking his return to the team with a goal, sending a towering header just wide on the half-hour mark.
A two-goal lead would not have flattered a United side who were putting in one of their best performances in recent weeks, but they then found themselves level with just two minutes remaining of the half as De Gea made another howler to gift Chelsea an equaliser.
Antonio Rudiger's 30-yard strike should have been simple for the keeper, but he spilled it into a dangerous area and Alonso was following up to convert the rebound - a third error leading directly to a goal from De Gea's last four games, which is as many as he committed in his previous 123 outings for the club.
Kepa Arrizabalaga showed his compatriot how it should be done when collecting a long-range Marcus Rashford free kick five minutes into the second half, and the Chelsea keeper was called upon again shortly afterwards to keep out Bailly's tame effort.
Clear chances were at a premium in the second half, though, and Chelsea in particular did not do enough to test the confidence of De Gea, whose only save until the final action of the match was a routine one to keep out Willian's shot at the front post.
Willian then sent a free kick over the crossbar before finding himself on the end of a poor challenge from Marcos Rojo, who was fortunate to only see yellow.
The challenge eventually forced Willian off the field as the Brazilian became the fourth player in the second half to limp off, with Rudiger, Rashford and Bailly also making way - the latter of whom appeared to sustain a particularly serious problem.
Those injuries resulted in seven minutes of added time at the end of the match, and Rojo almost took full advantage of his disciplinary reprieve to snatch the points for United when his header from a corner was cleared off the line by Pedro.
United did not do enough to push for victory throughout the second half, though, and they were almost left with nothing when Gonzalo Higuain was played through by Pedro in minute eight of stoppage time, only for De Gea to go some way towards redeeming himself by staying big to make the save.
Even so, a draw does little for United's Champions League ambitions ahead of their final two games of the campaign as another weekend passed without any of the top-four contenders winning.
MAN UTD (4-3-1-2): De Gea; Young, Bailly (Rojo 71'), Lindelof, Shaw; Herrera, Matic, Pogba; Mata (McTominay 81'); Lukaku, Rashford (Sanchez 65')
CHELSEA (4-3-3): Kepa; Azpilicueta, Rudiger (Christensen 65'), Luiz, Alonso; Kante, Jorginho, Kovacic (Loftus-Cheek 76'); Willian (Pedro 84'), Higuain, Hazard
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