Man United capitalised on two LFC first half defensive mistakes to take the advantage in the 170th league meeting of the clubs.
Marcus Rashford scored both goals whilst outfoxing Trent Alexander-Arnold, and Lovren was a passenger in aerial challenges with Romelu Lukaku.
The reds dominated the second half and had a number of shouts for penalties. A deep cross by Sadio Mane was put in to his own goal by Eric Bailly but the reds, despite a number of attacking substitutions could not add a second goal to bring back a point.
Henderson left out
Liverpool made 5 changes from the midweek Porto game, but compared to the Premier League clash last weekend there was only one difference - James Milner in for Jordan Henderson.
The reds dominated the opening ten minutes as United had men behind the ball but their counter attacking tactics got them two goals in the 14th and 24th minute of the match.
From a David de Gea kick Lukaku won the header and Rashford got behind Trent AA. He then turned the young Liverpudlian in the area before drilling a fine shot into the goal.
Ten minutes later, another de Gea launch. Lukaku won it and this time, via Mata, the ball found its way into the path of Rashford who's shot took a deflection off Alexander-Arnold and headed into the goal.
Second half improvement
The reds did improve, and looked fitter and stronger in the second half, but they had to chase the game.
Eric Bailly helped the cause by inexplicably putting the ball into his own net from a Sadio Mane cross. Mo Salah wasn't having much joy and in fact it was in the 95th minute when he almost scored the equaliser but drilled over the bar.
Liverpool had shouts for penalties on numerous occasions. Ashley Young held Salah in the area, there was a handball from an Andy Robertson cross and then Sadio Mane was clipped on the back of his heel in a 50/50 with Marouanne Fellaini.
Klopp took off both his full backs, bringing on Wijnaldum and Solanke but the equaliser would not arrive and United take a 5 points lead over Liverpool with 8 Premier League games to go.
Both clubs are a long way behind Manchester City who will win this seasons Premier League, but bragging rights for now rest with Mourinho.