Philippe Coutinho sent the Reds ahead in the sixth minute with a sublime curling shot inside the box after he was setup by Mohamed Salah, 1-0.
Andrew Robertson was not far from doubling the hosts' lead in the 16th minute with a powerful left-footed strike that sailed by inches wide of the upright.
Oliver McBurnie almost made the visitors' pressure count in the 37th minute after firing a dipping shot from just outside the box but it sailed over the crossbar.
Salah could have won the Reds a penalty on the stroke of halftime after being tripped by Alfie Mawson but the referee was not convinced.
The first stanza ended with Klopp's troops leading 1-0.
It took five minutes after the break for the Reds to find the cushion goal as Firmino got to the end of Coutinho's delightful free-kick and sent a composed finish past Lukasz Fabianski, 2-0.
Coutinho almost produced a moment of brilliance with a clever taken free-kick in the 56th minute that setup Salah inside the box, but the Egyptian's strike could not find the target.
10 minutes later, Trent Alexander-Arnold made it 3-0 with a scorching strike from just outside the box that found the roof of the net.
Firmino added the fourth goal in the 66th minute with a simple finish into an empty net after some good work by Salah in the build up.
Goalscorer Alexander-Arnold made way for James Milner in the last quarter of the match, while Leroy Fer got a knock in the last 10 minutes and was replaced by Renato Sanches.
Alex Oxlade-Chamberlain rounded off the scoring seven minutes before time with an impressive volley from close range, 5-0.