Ramires gave the reigning champion Chelsea an early lead before Brazil star Coutinho struck twice with Christian Benteke adding a late third for the visitors at Stamford Bridge.
A poor start to the EPL season, hardly helped by lackluster displays in the Champions League and a midweek League Cup exit to Stoke, is rapidly developing into a full blown crisis, with speculation about Mourinho's future rife.
The Chelsea faithful chanted his name as its team slipped to a sixth league defeat in just 11 games and the "Special One" was typically defiant in his post match press conferences.
'I have nothing to say'" he said repeatedly when pressed by broadcaster BT Sport about his future and later dismissed the suggestion that it was his last game in charge.
He also defended his players, largely unchanged from the squad which won the league title by eight points last season.
'If you saw some individual performances below a normal level -- there were not many. Two at maximum," he said.
"The players tried. You could feel the attitude. You could feel the desire and that the game was 50/50," was his verdict.
Most impartial observers saw it differently, aside from a bright opening by Chelsea as Ramires connected with a Cesar Azpilicueta cross to head a fourth minute goal.
Klopp's energized team took control thereafter, with Coutinho's equalizer in first half injury time befitting of his nickname of O Magico, curling a shot past Asmir Begovic from the edge of the area.
Chelsea's own talisman Eden Hazard was substituted early in the second half and Oscar nearly gave the home side the lead again with an audacious chip.
But it would have been against the run of play and Coutinho struck again in the 74th minute, his shot taking a slight deflection off John Terry to beat Begovic.
Klopp, who ironically labeled himself the "Normal One" in his first press conference after taking charge of Liverpool following the sacking of Brendan Rodgers, does appear to be working his own special brand of magic with his new team unbeaten in five games since he took over.
Confirmation of the victory came as Benteke shot through a despairing Gary Cahill with seven minutes remaining to grab the third in the 3-1 victory..
"It wasn't a good start but we stayed cool so I knew we would get some chances. We can do better but for today it was absolutely ok'," said former Borussia Dortmund manager Klopp.
Praise too for Coutinho, whose individual contribution proved the difference.
"Philippe is a great player," the German told BT Sport. "He has unbelievable skill, he's a great guy."
Man City top
Later Saturday, Manchester City shrugged off a sorry mistake by its goalkeeper Joe Hart to beat Norwich City 2-1 to stay top of the standings.
Hart blundered to allow Cameron Jerome to equalize Nicolas Otamendi's opener for City, but Yaya Toure put the home side ahead again from the spot after Russell Martin's handball.
There was still time for Aleksandar Kolarov to miss a further spot kick for laboring City.
Arsenal is level on points with City after an impressive 3-0 win at Swansea City, while Manchester United lost ground with a goalless draw at Crystal Palace. It allowed Leicester City, 3-2 winners at West Bromwich Albion with Jamie Vardy scoring for an eighth straight game, to move into third.
Real and Barca win
In La Liga, it was a case of no change at the top as both Real Madrid and Barcelona claimed comfortable wins Saturday.
Real Madrid beat Las Palmas 3-1 in the Bernabeu as Isco, Cristiano Ronaldo and Jese Rodriguez scored.
But Rafa Benitez's injury-hit squad suffered another blow when Luka Modric limped off and is doubtful for the midweek Champions League clash with Paris Saint Germain.
Real tops Barcelona on goal difference, the Catalans drawing level on points with a later 2-0 win at Getafe with Luis Suarez and Neymar on target.
Both were on assists by Sergi Roberto, the first a stunning and impudent back heel for Suarez to convert easily, the second a pin point pass for Neymar to volley his ninth La Liga goal of the season.