The visitors fell behind to Ramires' early header but fought back to claim a deserved three points, with Philippe Coutinho scoring twice and Christian Benteke adding a late third.
Jurgen Klopp celebrated the win - his first in the Premier League since replacingBrendan Rodgers - wildly but his managerial counterpart Mourinho wore the look of man who knows his time could be up.
Chelsea have won just three of their 11 league games this season and are 15th in the table following their latest defeat.
It had started so well for Mourinho's men against Liverpool as they made the breakthrough inside four minutes.
Ramires had not scored at Stamford Bridge since December 2013, but the Brazilian showed a striker’s instinct to hold off Alberto Moreno and head home Cesar Azpilicueta’s pinpoint cross.
Chelsea failed to build on their early breakthrough, however, and Liverpool enjoyed the majority of possession as they grew into the game.
Coutinho, operating behind frontman Roberto Firmino, had looked their biggest threat throughout the first-half and it was the tricky Brazilian who pulled the vistiors level two minutes into injury time.
Receiving the ball on the edge of the box, Coutinho cut inside Ramires and curled a delightful left-footed shot past the despairing dive of Asmir Begovic.
Liverpool resumed the second half where they had left off and continued to put the hosts, who looked nervous throughout, under pressure.
The anonymous Eden Hazard was replaced with Kenedy as Mourinho desperately attempted to spark his side into life.
But it failed to have an effect and Liverpool's pressure finally told as Coutinho claimed his second when his from the edge of the area deflected off John Terry and past Begovic.
Subsitute Benteke then added to Chelsea's misery as he struck a third after Gary Cahill and Terry had backed off.
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