Both sides had half-chances in the early exchanges but it was Suarez who gave the home side the lead mid-way through the opening 45 after directing Jordi Alba's slick pass into the back of the net.
Despite being under pressure for much of the half, it was Barcelona who scored against the run of play to double their advantage with a lucky deflection off the crossbar falling into the path of Messi to convert.
With the Reds looking to score an important away goal, Ernesto Valverde's side began to create more chances on the counter but it was a free-kick that eventually saw them add a third goal.
There was little Liverpool could do as Messi stepped up to smash home a sensational free-kick past the fingertips of Alisson from a little over 25 yards out.
The result was a huge blow for Liverpool, who would be right to feel hard done-by after controlling much of the game and penning Barca back.
Jurgen Klopp's side have a huge task ahead of them at Anfield in six days time, with a crucial Premier League fixture against Newcastle to come before that.
Slide through our gallery below for the best snaps from a disappointing evening for the Reds.