Two goals down inside 12 minutes, the leaders fought their way off the ropes to level through Juan Mata (44) and Ramires (53) and were looking likelier winners until Branislav Ivanovic was sent off on the hour.
But a second red card for Fernando Torres meant the end of any genuine attacking ambition, and substitute Javier Hernandez bundled a contentious winner (75) to haul United within a point of the leaders and give them a first league success at Stamford Bridge since 2002.
Despite the result there was plenty in Chelsea’s performance to reinforce their credentials as real title challengers, though they were handicapped by a haphazard start as United’s £24m man Robin van Persie made one goal and scored another.
United’s defence was under scrutiny before the game following another porous display against Braga in midweek, but it was a home back four missing John Terry which was exposed time and again during the opening exchanges.
David Luiz was too often in a different district to his fellow centre-half Gary Cahill while Ashley Cole was repeatedly isolated on the left flank, from where both early goals came.
Ashley Young strolled into the gaping hole between Luiz and Cahill to feed Wayne Rooney, and when his cut-back was met by a firm first-time van Persie shot the ball hit a post, cannoned off Luiz and bobbled beyond Petr Cech.
Van Persie’s movement in the box for the opener was bright enough, but it was an identical checked run which caught out Cahill soon afterwards and this time the striker’s right-footed effort from Antonio Valencia’s low centre beat Cech by itself.
Luiz was slow to close down Valencia and continued to unsettle home supporters at the back, but it was the Brazilian’s free-kick which forced David De Gea into the first of several feet-first saves and sparked an improvement in Chelsea’s performance.
Jonny Evans miscued a clearance against a post, De Gea denying Cahill from the corner which followed, and the goalkeeper then used his right hand to claw out a Torres header from his near post.
A goal was coming and Mata’s precise free-kick left his compatriot little chance before Ramires rose above Tom Cleverley to head a second-half equaliser.
Eden Hazard then went close as Chelsea threatened to become the first side in Premier League history to topple United from 2-0 down, but Ivanovic saw red for clipping a clean-through Young and Torres followed for a second booking – referee Mark Clattenburg deciding he dived unnecessarily over Evans’ lunge.
Chelsea were left hanging on and the task proved too much as Cech kept out van Persie only for Hernandez – loitering in what appeared an offside position – to turn in Rafael da Silva’s follow-up.