Firmino then made it 3-1 to give Liverpool a deserved foothold in the game before a Marcus Rashford strike set up a nervy ending for the Reds. But Mohamed Salah put the icing on the cake after a quick break and finish to give Liverpool all three points.
The result sees them move up to fifth in the table - four points behind Chelsea having played a game less and with their top-four destiny still in their own hands.
It was an open start to the game with both sides looking to get on the front foot from the off. But it was the hosts that nudged ahead inside ten minutes as Bruno Fernandes slotted into the back of the net.
Aaron Wan-Bissaka, who got to the byline, laid the ball back for Fernandes, whose outside-of-the-foot effort hit Nathaniel Phillip’s outstretched leg and into the opposite corner past Alisson.
Liverpool thought they had a lifeline midway through the half as Eric Bailly caught Phillips just inside the box, but referee Anthony Taylor overruled his original decision after taking a look at VAR - much to Klopp’s bemusement on the sidelines.
But the Reds found a leveler soon after as they began to build up pressure around United’s box. Jota found himself on the end of Phillips’ wayward shot after the centre-back won the ball back inside the box, with Jota cleverly back-heeling the ball into the back of the net past Dean Henderson to make it 1-1.
Liverpool looked dangerous going forwards as they looked for a second, although United continued to pose a threat themselves as they looked for pockets of space in behind the inexperienced duo of Phillips and Rhys Williams.
The Reds then completed the turnaround before half-time as Firmino got on the end of Trent Alexander-Arnold’s exquisite set-piece and headed home to make it 2-1 at the break.
Liverpool then made the perfect start to the second half as Alexander-Arnold’s strike was parried by Henderson, with Firmino tucking home the rebound to give Liverpool a 3-1 lead.
The hosts were unable to mount a response as Liverpool impressively stifled their attacks and looked favourites to add a fourth goal.
Good play between Salah and Firmino saw the Brazilian play Jota through on goal, however a heavy first touch narrowed the angle with the Reds forward only able to direct his effort against the woodwork.
Liverpool had United on the ropes as a counter-attack from a United set-piece saw Alexander-Arnold with space down the left. However, his left-footed finish across the face of the goal was kept out by Henderson.
The introduction of Mason Greenwood gave Ole Gunnar Solskjaer’s side encouragement and the hosts pulled one back soon after as Edinson Cavani fed Marcus Rashford, who made no mistake with a toe-poke finish past Alisson to make it 2-3.
Greenwood nearly grabbed an equaliser but his close-range effort was cleared off the line by Phillips as Liverpool eventually squashed the danger by conceding a corner which resulted in nothing of note.
It was a nervy ending for the Reds but it was settled in the final minutes by Salah, who ran through on goal after being found by Jones before finishing past Henderson to give Liverpool all three points.
Another three wins out of three sees Liverpool clinch a top-four spot - which is a remarkable feat given their horrid run at the start of this calendar year.
MOTM: Trent Alexander-Arnold
Next Fixture: West Brom (A) - 16 May, 16:30 BST