Philippe Coutinho scored two fine goals after Ramires had given Chelsea a fourth-minute lead to pile more pressure on Mourinho, who only in May won his third Premier League title.
And Christian Benteke's third allowed Jurgen Klopp to celebrate a second victory in his fifth game as Liverpool boss since replacing Brendan Rodgers.
It left Mourinho pondering his future and further perceived injustices at Stamford Bridge.
Klopp and Mourinho each berated the officials from their technical areas, the Chelsea boss treading carefully given his recent disciplinary run-ins with the Football Association.
Mourinho had the most legitimate complaint, though, when Lucas avoided a second yellow card for a challenge on Ramires with more than 20 minutes to go and the score 1-1.
Coutinho's second followed and Mourinho tried to lift his team amid chants of "You're getting sacked in the morning" from the vocal visiting support, who were able to enjoy Benteke's strike.
Chelsea now have one win in eight games and Mourinho, who was subject to the first managerial vote of confidence in Roman Abramovich's 12-year ownership on October 5, will be wary of what the future holds.
Abramovich was absent from Stamford Bridge so did not hear the defiant and supportive chants from the home faithful as Chelsea's season slipped deeper into crisis.
Diego Costa passed a late fitness test on a rib injury to start for Chelsea, who made a fine start when Ramires headed in Cesar Azpilicueta's cross.
Ramires this week extended his Chelsea contract until June 2019 - in line with his manager's.
Liverpool were coming more and more into the game, but the transition from defence into attack was not swift enough for Klopp's liking
Adam Lallana stretched to reach a James Milner cross but could only shoot straight at Asmir Begovic and Klopp was frustrated John Terry was not penalised for handball
It would have been harsh.
Referee Mark Clattenburg also gave Emre Can only a ticking off, rather than a caution, for a reckless lunge on Willian.
Chelsea were playing on the break, with Liverpool the more inventive.
Lucas had a header saved from Milner's cross before, in added time, Coutinho's body swerve wrong footed Ramires and his left-footed curling shot soared in.
Chelsea heads dropped and Mourinho had to lift them at the interval, but Liverpool resumed where they had left off.
Gary Cahill was subject of a muted handball appeal and shakily conceded a corner.
Klopp complained repeatedly about decisions to fourth official Lee Mason and Mourinho pointed out his recent disciplinary run-ins.
Kenedy replaced the ineffective Eden Hazard and immediately tore through and shot narrowly wide.
Klopp turned to Benteke, but might have been down to 10 men only for Lucas, who had been booked earlier in the half for persistent fouling, to avoid a second caution for a challenge on Ramires.
Mourinho sarcastically applauded the decision which brought Chelsea's entire bench to their feet.
Mignolet back-pedalled to save Oscar's effort from near halfway, but it was Coutinho who struck once more as his shot from the edge of the area found the corner.
Begovic saved with his feet to deny Alberto Moreno as Liverpool came close to a third, which came when Benteke struck as Cahill and Terry backed off.
Source : PA