Frank Lampard's side were punished for some dismal defending at Bramall Lane as David McGoldrick struck twice, either side of Oli McBurnie's goal.
The Blues remain third after a second defeat in their last four games.
They will be down to fifth place by Monday evening if Leicester beat Bournemouth and Manchester United defeat Southampton.
It was another alarmingly sloppy display from Chelsea, whose other defeat since the restart was a defensively shambolic 3-2 loss at lowly West Ham.
With games against Norwich, Liverpool and Wolves left in the league, Chelsea will hope second-placed Manchester City's two-year European ban is upheld on Monday.
In that case, finishing fifth would guarantee Champions League qualification.
The Blades are now unbeaten in their last four games, with Chelsea their latest scalp after impressive wins over Tottenham and Wolves.
Chris Wilder's team are up to sixth place and are firmly in the hunt to qualify for the Europa League at the end of a remarkable first season back in the top flight.
Lampard made two changes as Jorginho came in for Billy Gilmour, who had knee surgery after Tuesday's win at Crystal Palace, and Tammy Abraham started in place of Olivier Giroud.
In contrast with what was to follow, Chelsea made a bright start and Ross Barkley tested Dean Henderson before Christian Pulisic headed over from Reece James' cross.
It was United who took the lead in the 18th minute as their high-tempo pressing force a naive mistake from Mason Mount.
The Chelsea youngster was caught in possession as he tried to dribble out of his own penalty area and George Baldock crossed back into the box.
McBurnie met the delivery with a volley that deflected off Barkley, forcing Kepa Arrizabalaga into a awkward save that only pushed the ball out to McGoldrick and he slotted home from close range for his first Premier League goal.
Conceding such a messy goal knocked Chelsea out of their stride and the Blades doubled the lead in the 33rd minute after more sloppy defending from the visitors.
The fragility of Chelsea's defence has been their fatal flaw several times this season and they were prised open far too easily when McBurnie got in front of Kurt Zouma to meet Enda Stevens' cross with a powerful header past Kepa.
James' long-range effort forced a good save from Henderson, but the visibly frustrated Lampard replaced Mount and Andreas Christensen with Marcos Alonso and Antonio Rudiger at half-time.
Abraham should have given Chelsea a lifeline midway through the second half, but the striker poked over.
McGoldrick put Chelsea out of their misery in the 77th minute.
He started the move on the halfway line and played a perfectly-weighted pass to Lys Mousset.
Mousset cut it back into the penalty area and when Rudiger made a complete hash of clearing, McGoldrick was on hand to blast into the roof of the net.