The Reds took to Goodison Park for their 227th clash with their city rivals, desperate to keep up the pace at the top end of the Premier League.
With Chelsea securing their 11th straight victory on Saturday afternoon, Liverpool needed to take a victory back with them as they made the short trip across Stanley Park.
The clash brought Klopp’s first taste of Goodison, and Ronald Koeman’s first taste of the derby.
This was the case all over the field, with Everton looking to rattle the visitors with a fierce, high-tempo start.
Divock Origi toiled away in Firmino’s place as Liverpool’s centre-forward, but snatched at a series of chances.
With the game balanced, Mike Dean blew for half-time with the scores settled at a disappointing 0-0.
Tensions flared early on in the second half as Ross Barkley slid in with a vicious challenge on Jordan Henderson, with Lovren getting embroiled in an argument with Romelu Lukaku.
Barkley somehow avoided a red card, though Koeman was adamant after the game that Lovren should have been given a second yellow for his involvement.
It was Sturridge’s intervention that crafted Liverpool’s goal, with Mane following up on the striker’s rebounded effort to tap home from close range and seal the three points.
This sent the travelling Kop into raptures, as Liverpool recorded their first 1-0 victory of 2016.
There were more celebrations at full-time, with Klopp and his players now set for their Christmas party on Tuesday, after reducing the gap between them and Chelsea to six points.