Matthew Pennington equalised for Everton after Mane's early opener, but otherwise the hosts largely controlled the game.
Nonetheless, it would soon be 3-1 when the brilliant Coutinho again ran at Everton down the left side, and then fed in substitute Origi who fired home from the edge of the penalty area with Robles wrong-footed in the 60th minute.
Speaking after his side's victory, Jürgen Klopp, the Liverpool manager, suggested that Mane's injury was not too serious but believes he will not be available for Wednesday's match at Anfield. "It was Roberto all over the pitch", Klopp added, insisting he did a "really good job". We did really well, scored fantastic goals.
"We had to defend at the end and our organisation was not that good, but we defended with passion and could have had one or two more clear counter attacks, but all good".
"Today we played a really good game and almost everything was ideal, and then we lose Sadio Mane", added Klopp.
"We are different to them".
"We made it really hard for Liverpool". We came back after an unlucky goal between the legs.
"I thought it was deserved but like always you have to learn a bit from the game", said the German.
Before the game there was a tribute to Ronnie Moran - this game being the first since the legend of Liverpool passed away. It started with our mistakes in both positions and they punished us for this.
The Everton goalkeeper was nearly further embarrassed when he palmed Coutinho's shot up and onto the crossbar, a warning Everton did not heed when the Brazil global once again cut back onto his right foot to curl a shot inside the same post on the half-hour. His goal demonstrated fine dribbling and finishing ability.
Liverpool handled the occasion better than its rival. His big players let him down. "It is not the first time". If you saw something - say it. And bad news was to follow when Sadio Mane appeared to be seriously struggling with an ankle injury and was withdrawn for Divock Origi.
In a foreshadowing of the second goal, Coutinho went on surging run of his own in which he wrong-footed Pennington but Robles got a strong palm to the Brazilian's shot and Phil Jagielka headed off the goal line.
Koeman brought on Enner Valencia and Gareth Barry, but Mason Holgate was the next player to threaten, having a deflected effort saved by Mignolet and heading off target from Baines' cross a few moments later.