Liverpool had been second best throughout their defeat to Leicester City on Monday night, but Jurgen Klopp's side were positive from the opening whistle against Arsenal and deservedly went in front after nine minutes.
A long ball found its way to Adam Lallana and after he gave Mane the chance to cross from the right, he found Firmino at the back post who fired into the roof of the net.
That remained the only chance during the first quarter of the contest but as Arsenal struggled to create anything in the final third, Liverpool were building up a head of steam.
Just before the half-hour mark, Philippe Coutinho's strike was tipped over the crossbar by Petr Cech, and it was not long before the home side found a second goal through Mane, who hit the bottom corner from 12 yards after a fine move involving James Milner, Wijnaldum and Firmino.
Just before half time, Coutinho failed to lift the ball over Cech from 12 yards after a wonderful piece of control and after the restart, he was almost made to pay for failing to extend his team's lead.
Alexis Sanchez had been introduced at half time and three minutes after his introduction, his cross into the penalty area found Olivier Giroud who saw his header tipped onto the crossbar by Simon Mignolet.
Liverpool continued to press but 12 minutes into the second half, Arsenal halved the deficit through Danny Welbeck as he clipped the ball over Mignolet after collecting Sanchez's pass.
The home side were rocked but to their credit, they remained on the offensive with both Nathaniel Clyne and Coutinho shooting over from distance and Joel Matip wasting an headed opportunity at the back post.
As the game entered the final 10 minutes, Divock Origi struck the post with a header as the home side continued to threaten a third before they finally wrapped up the game in added-on time when Wijnaldum finished off a sweeping break involving Lallana and Origi with a calm finish into the bottom corner. body check tags ::