Ramires put the hosts ahead in the fourth minute at Stamford Bridge, but Coutinho equalised in first-half stoppage time and put Liverpool ahead with a deflected shot in the 74th minute before substitute Christian Benteke rubber-stamped Chelsea's sixth defeat in 11 league games.
While visiting manager Jurgen Klopp was left to celebrate his first league win -- Liverpool's first on the road since the season's opening weekend -- it was a Halloween nightmare for Mourinho, whose job is thought to be on the line after a run of just one win in eight matches in all competitions.
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Some newspaper reports had even suggested that he could be dismissed in the event of another defeat and he was assailed with chants of "You're getting sacked in the morning!" by the away fans as the game wound down.
Mourinho made just one change to the team eliminated from the League Cup by Stoke City in mid-week, with Cesar Azpilicueta replacing Baba Rahman, which meant that Cesc Fabregas and Nemanja Matic remained on the bench.
Diego Costa passed a fitness test after bruising a rib at the Britannia Stadium and he wasted no time in making an impact.
The Spain striker's back-heel on the left-hand touchline freed Azpilicueta to cross from the left and Ramires stole in front of Alberto Moreno to celebrate his new four-year contract by heading in the opening goal.
It was the kind of start that has eluded Mourinho of late and it allowed Chelsea to retreat into their own territory and play on the break.
The right flank proved fruitful territory for the visitors, with full-back Nathaniel Clyne and James Milner doubling up on Azpilicueta, but although Adam Lallana, twice, and Lucas Leiva hit the target, Chelsea goalkeeper Asmir Begovic was not troubled.
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However, with first-half stoppage time having ticked 30 seconds beyond the allotted two minutes, Liverpool hit back in style, Coutinho wrong-footing Brazil team-mate Ramires and curling an unerring left-foot shot around Begovic from the edge of the box for his first goal in 13 appearances.
A cagey opening to the second period saw tempers flare in the dug-outs, with Mourinho drawing fourth official Lee Mason's attention to Klopp's quasi-maniacal appeals before the German became involved in an angry exchange with a member of Chelsea's back-room staff.
Midway through the half, Mourinho leapt to his feet in disbelief when Lucas avoided a second booking for a trip on Ramires, demonstrating his displeasure with sarcastic applause for referee Mark Clattenburg.
Mourinho hooked the disappointing Eden Hazard in the 58th minute, with Kenedy coming on, while Benteke replaced Milner as Liverpool switched to a 4-2-3-1 system.
Within two minutes Kenedy had shown more enterprise than Hazard had in an hour, cutting in from the left and shooting wide.
Oscar also chanced his arm with an audacious 45-yard lob that Simon Mignolet had to stretch to claw away, but it was Liverpool who struck next.
A high ball was headed down by Benteke and Coutinho steadied himself before dispatching a right-foot shot that flicked off John Terry and flew past Begovic.
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After Moreno and Jordon Ibe had gone close, Benteke made it 3-1 in the 83rd minute, gathering Ibe's pass and drilling a low shot into the bottom-left corner to seal Liverpool's first win at Chelsea since November 2011.
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