A lightening start saw Kevin Strootman test Loris Karius from distance, while Salah did the same at the other end against Alisson.
Liverpool managed to get in behind the Roma backline and Roberto Firmino flashed a chance across goal, before Alex Oxlade-Chamberlain tried his luck from outside the area.
Hearts were in the Liverpool fans' mouths as Aleksandar Kolarov smashed the crossbar with an effort from 25 yards.
From there, Liverpool dominated and had Roma on the back foot for the remainder of the half, with Sadio Mane missing a trio of excellent chances.
First blasting one over when one on one with Alisson, then after a fine cutback putting into the stands from 12 yard, and finally he did get the ball in the net but he was adjudged to be in an offside position.
Alisson had to be alert to the threat of Salah, first making a save from the Egyptian, then denying Roberto Firmino.
However, that was just a warning as Salah picked up the ball on the edge of the penalty area and curled a brilliant shot into the far top corner.
Things almost went from bad to worse as Dejan Lovren headed onto the crossbar from a corner, then Alisson once again denied Mane.
But just as half-time approached, a quick counter attack from Liverpool saw Salah run in on goal, and dink the ball over Alisson for his second.
Roma changed things around at the start of the second period, Patrik Schick coming on for Cengiz Under, however, it was the hosts that remained on the front foot.
Roberto Firmino's excellent hold-up play was allowing Salah to find the gaps that kept appearing between Aleksandar Kolarov and his midfield.
The hero against Barcelona in the last round, Kostas Manolas was at least putting himself about for the Italians, but it was scant consolation for a team that was getting completely overrun.
Mane was flagged offside on 53 when well placed, with the chance coming about because of another error in Roma's defence.
Even long balls from the back were causing the Italians to panic.
Within two minutes, Salah turned provider.
Wide open on the left-hand side with the visitors playing a high line, he looked for Mane who stroked it home.
Roared on by a vocal Kop, Liverpool just kept on coming.
Salah again turned provider on 61, this time sliding in Firmino who couldn't miss from two yards out.
Roma responded by screaming at their coach Eusebio di Francesco, as if to say 'what on earth do we do.'
Even the normally reliable Alisson dropped a straightforward catch as the visitors continued to crumble under incessant pressure.
Far from dropping off, Liverpool continued to pour forward, but their fifth goal was a straightforward header from a corner, Firmino getting his second on the night with 22 minutes still left to play.
Roma's Perotti almost pulled one back on 72 with a towering header, but it went straight into Loris Karius' grateful arms.
On 80 minutes, and out of nothing, Dzeko managed to score a crucial away goal - just as he had at Barcelona, and which proved crucial in Roma progressing to this stage.
Immediately thereafter, Georginio Wijnaldum completely missed his kick in front goal, allowing the visitors to break again; this time Dzeko's shot was far too weak.
With renewed impetus, Roma started playing again, and won a penalty for a James Milner handball with six minutes to go.
Perotti dispatched it into the roof of the net and all of a sudden there was a completely different complexion on a game that had seemed dead and buried five minutes before.
Liverpool's game management had gone out of the window, as it had against West Brom a few days earlier, and from a position of strength, they spent the final few minutes holding on to a three-goal advantage; something that Roma managed to reverse against the Catalans.
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