The Catalans were not at their sparkling best but still proved far too good for Roma, who were unfortunate to score two own goals through Daniele De Rossi and Kostas Manolas before Gerard Pique added a third.
Edin Dzeko pulled one back in the 80th minute to keep Roma just about in the tie but Suarez netted three minutes from time to put Barca within sight of a first semi-final appearance in the competition for three seasons.
Barca boss Ernesto Valverde sprang a surprise in his team selection by leaving both Paulinho and Ousmane Dembele out, with Nelson Semedo making the starting line-up. Radja Nainggolan failed to recover from injury in time for Roma so Alessandro Florenzi started on the right of the front three.
Roma set their stall out in the first minute, pressing high and snapping into tackles. A Diego Perotti cross caused panic in the Barcelona defence but Jordi Alba was able to clear with Florenzi lurking.
At the other end, Suarez had the ball in the net in the seventh minute after running on to a defence-splitting pass from Ivan Rakitic but the flag was raised for offside.
Roma were closing down Lionel Messi in numbers but still the Argentinian engineered a chance in the 11th minute, turning and firing in a shot that Alisson pushed away.
Barca came even closer seven minutes later when the lively Rakitic curled a shot from the left-hand edge of the box that hit the far post.
The hosts continued to threaten, with Suarez the next to go close. Messi’s back-heel played in the Uruguayan and his low shot was expertly pushed around the post by Alisson.
When the goal did come, it was a stroke of ill fortune for Roma. Andres Iniesta attempted to pick out Messi and, in stretching to cut out the pass, the impressive De Rossi inadvertently applied a finish worthy of the Barca magician.
Roma responded well and, had Samuel Umtiti’s foul on Lorenzo Pellegrini come a second later, he would have conceded a penalty. As it was, the visitors had a free-kick on the edge of the box but Pellegrini’s effort was blocked by the wall.
The visitors were still looking for a clear-cut chance and they got one within seconds of the restart. Florenzi’s superb cross beat the Barca defence but Perotti headed badly wide when he should have found the net.
It was a miss they would quickly regret as Barca began to lay siege to Alisson’s goal. The keeper fumbled a Jordi Alba shot only for Suarez to poke the rebound over, while Messi then shot straight at the Brazil international.
But they doubled their lead 10 minutes after the break when Messi and Rakitic engineered a short corner and the Croatian sent in a cross. Umtiti’s shot hit a post but the rebound went in off Manolas.
Alisson was called on again to keep out a Rakitic effort moments later but, after saving from Suarez in the 59th minute, was beaten for a third time when Pique tapped in the rebound.
Roma needed something to give themselves a sniff in the second leg and, after Marc-Andre ter Stegen had denied substitute Gregoire Defrel and Perotti, Dzeko applied the finishing touch.
But their hard work was undone seven minutes later when poor defending was emphatically punished by Suarez.