United were heading to what would have been a morale-boosting three points, thanks to a brilliant turn and half-volley finish from Jesse Lingard in the 61st minute.
It was tough on United, who belied the criticism of their play as sterile under manager Louis van Gaal, putting in a performance brimming with attacking intent.
United were particularly adventurous down the flanks, with full-backs Cameron Borthwick-Jackson and Matteo Darmian providing width and the former getting forward to set up Lingard’s opener.
Anthony Martial came close to giving United the lead before Lingard’s strike, only for Thibaut Courtois to acrobatically keep out his curling sho.t
Both goalkeepers demonstrated their pedigree and, without custodians of the quality of De Gea and Courtois, it could conceivably have ended up 3-3, rather than 1-1.
A draw may have been disheartening for Van Gaal’s team, who forced 10 corners in the first half alone, but Chelsea deserved credit for responding to going behind by turning the tide and ramping up the pressure on United.
With time running out, the reigning champions began to force the Red Devils back as they monopolised possession of the ball.
Van Gaal left the pitch in discussion with the match officials, but upon reflection he will surely be angrier with the manner in which his team allowed Fabregas to thread the ball through a crowded, central area of the pitch, into Costa’s path in the penalty box.
Daley Blind in particular seemed to go walkabout, then crucially slipped as Costa evaded the United defence, a minute into the six minutes added on at the end of the second half.
The result means Guus Hiddink’s Premier League record in two spells with Chelsea reads 20 wins, 11 draws and just one defeat, while Van Gaal has overseen just a single win in nine matches against sides managed by his Dutch compatriot, stretching back to 1998 (Hiddink has won four).
It’s not all positive for Chelsea, though, as they have now gone four Premier League matches at Stamford Bridge without victory, and remain 13th in the Premier League table, still looking down rather than up.
Hiddink also saw Kurt Zouma stretchered off in agony with what appeared to be a serious knee injury, after landing awkwardly just before the hour mark.
The Chelsea defender’s injury was a low point in an otherwise entertaining encounter that hinted at brighter times ahead for two clubs struggling under the weight of expectation this season.
Where they go from here will, like this encounter, be intriguing to see.