Chelsea went into the match after another protest against the club's ownership, with more than 500 supporters chanting against the regime as they marched alongside fans of French club Strasbourg, part of the same multi-club group.
On the pitch, Chelsea started brightly but soon ran into an increasing problem: scoring goals.
This latest defeat means it has been almost six and a half hours since Chelsea last scored, in the 4-1 win at Aston Villa.
In the Premier League, Chelsea have lost five of their past six games, have not won at home in five matches and have won only one of their past eight matches.
It is their sixth home defeat in the league this season, including three in a row, and only in 1994-95 have they lost more games at Stamford Bridge.
Chelsea have failed to score in each of their past four defeats, their joint longest such barren run since 1912.
Regular starting striker Joao Pedro was ruled out before kick-off with a minor quadriceps injury, while Estevao being forced off compounded the difficulty.
Those called upon have struggled for form, and again failed to deliver in what was close to a must-win match for Chelsea if they were to maintain realistic hopes of qualifying for the Champions League.
Substitute striker Liam Delap extended his goal drought to 20 games, winger Pedro Neto is without a goal in nine matches across all competitions, Cole Palmer has now gone seven games without scoring, and Estevao's effort against League Two Port Vale two games ago remains his only recent goal. The teenager has not scored in the league since January.
His replacement, summer signing Alejandro Garnacho, who arrived from United for £40m, has not scored a league goal since October.
As well as the three efforts that hit the woodwork, Enzo Fernandez and Palmer wasted presentable chances.
What was once Chelsea's main strength – scoring goals – has become their biggest weakness at the most critical stage of the season.
There remains criticism of the club's goalkeepers and the quality of their defensive options, and questions about how the season has come to a halt when Liam Rosenior has had free midweeks to improve the club's fortunes on the pitch.
However, most of the anger has been directed at the ownership, despite co-owner Behdad Eghbali saying in a rare interview this week that mistakes had been made in recruitment and that more experience may be needed in the league's youngest squad.
Even star midfielder Moises Caicedo renewing his contract cannot detract from a bleak picture at Stamford Bridge.
Those chants of "we want our Chelsea back" went from the streets outside the stadium into the stands during the second half, and the team were booed off at the final whistle.
