James Milner doubled the advantage via a penalty before Thomas Meunier pulled one back for the visitors.
It was nearly 2-0 a minute later, PSG scrambling the ball away after excellent work from Mohamed Salah had created havoc, but Mbappe then missed a chance to level when he lifted over from inside the box.
Mohamed Salah thought he had restored the hosts' two-goal cushion on 58 minutes as he tapped home from close range after Areola spilled the ball from Sturridge's sliding challenge, but the referee blew the striker up for a foul.
The Belgian scored a quality goal in the first half which meant that Liverpool went into the break with the Paris-based club breathing down their necks.
Around six months ago, Daniel Sturridge was on-loan at West Brom and his Liverpool career looked over.
Klopp set his team out to dominate the midfield from start to finish, removing the dynamism of Naby Keita for the leadership and solidity of captain Jordan Henderson in the starting lineup, and this was a move that worked to perfection.
Yes, he is not the most gifted or glamorous midfielder in world football.
Star man Neymar was largely transparent in the final third and cut a frustrated figure for most of the match, with his downtrodden body language aligning itself with his considerable lack of effort to defend when losing possession.
Neymar attempted to find Kylian Mbappe inside the area as PSG tried to exert pressure, but they were back on the defensive when, from another Liverpool corner, Joe Gomez shot over.
"The problem for Liverpool is the fans don't get up for the league games in the same way as they do in the Champions League", the Fleetwood boss said.