The visiting side's failure to take their opportunities cost them as Messi scored twice in quick succession, the second a sublime free kick, to give Barcelona a huge advantage ahead of next Tuesday's return leg at Anfield.
Barcelona had two ex-Liverpool attackers in their starting ranks, meanwhile, and it was one of those who made the only major impact in the first half.
Jordi Alba played a fine pass in behind and Suarez made the most of some static defending to flick the ball past Alisson Becker with a deft touch inside the six-yard box, ending his year-long draught without a goal in this competition.
Philippe Coutinho, the other of Barca's ex-Liverpool men, also got a shot on target but Alisson was equal to his compatriot's trademark curler.
Liverpool more than held their own in the opening 45 minutes, however, and were a little unlucky to go into half time a goal behind.
After having an early penalty appeal rejected following a slight push from Gerard Pique, Sadio Mane guided a first-time shot over the crossbar after being picked out by Jordan Henderson, who was brought on for injured teammate Naby Keita.
The Reds, 2-1 winners at this ground on their most recent visit here 12 years ago, came close to a leveller early in the second half when James Milner's shot from 12 yards was pushed to safety by Ter Stegen.
Ter Stegen was required five minutes later to turn aside Mohamed Salah's long-ranger, which was heading for the bottom corner of the net, and he produced the third of his stops on the hour to keep out Milner.
Salah controlled the ball with a sublime touch down the right and Georginio Wijnaldum dummied the ball once it was cut back, but Milner's bullet shot was aimed too close to Ter Stegen.
Suarez got on the end of a deflected Messi through-ball and hit the crossbar with his shot, which Messi was well positioned to roll into an unguarded net from the rebound.
From effectively holding on to killing off the tie in the space of seven minutes, Barca had a third goal through a magical Messi free kick, sending the ball over the wall and right into the top corner of Alisson's net for his 600th career club goal.
Liverpool should have at least had an away goal to take into next week's return match, only for Salah to pick out the post from six yards after substitute Firmino's shot from close range was blocked on the line.
Barca had a couple of promising chances on the counter in the closing stages, though, the best of which fell to Ousmane Dembele from the final kick of the match - the Frenchman scuffing his shot straight at Alisson to leave Liverpool with just a glimmer of hope of another famous European comeback.
BARCELONA (4-3-3): Ter Stegen; Roberto (Alena 94'), Pique, Lenglet, Alba; Rakitic, Busquets Vidal; Messi, Suarez (Dembele 93'), Coutinho (Semedo 60')
LIVERPOOL (4-3-1-2): Alisson; Gomez, Matip, Van Dijk, Robertson; Milner (Origi 84'), Fabinho, Keita (Henderson 24'); Wijnaldum (Firmino 78'); Salah, Manebody check tags ::