Mohamed Salah almost opened the scoring for the Reds in the third minute after working his way inside the box, but his low left-footed strike was saved by Roma shot-stopper Alisson.
The Reds were dealt a blow in the 18th minute after Alex Oxlade-Chamberlain picked up a knee injury and was replaced by Georginio Wijnaldum.
Jurgen Klopp’s charges had not really settled after half-an-hour, owing to the earlier change and they could not create clear opportunities.
However, Salah put them ahead in the 36th minute with a moment of individual brilliance, as he sent a fine curling effort into the top corner from the edge of the box.
The Egyptian international made it 2-0 in the stroke of halftime by lifting a perfectly weighted dink over Alisson following a through ball from Roberto Firmino.
Ten minutes into the second stanza, Sadio Mane got the third goal for the Reds with a simple tap-in at the far post corner following a cross from Salah.
Firmino made it 4-0 on the hour-mark as Salah turned provider once again to tee up the Brazilian for the easy finish.
The fifth goal arrived in the 68th minute as Firmino rose above his marker to send a fine header to the far post to complete his brace.
With 10 minutes remaining, Eden Džeko got a crucial away goal for Roma with a clinical close-range strike.
Diego Perotti further reduced the deficit to 5-2 five minutes before time from the penalty spot to give his side a glimmer of hope heading into the second leg.