Neymar's early strike appeared to erase any forlorn hopes of an Arsenal comeback, but the visitors were given hope when Mohamed Elneny drilled into the top corner just after half-time.
Suarez put the tie beyond Wenger's men in emphatic fashion with a stunning volley 25 minutes from time before Messi's cheeky chipped finish rounded off the scoring late on.
A ninth consecutive victory stretches Barca's unbeaten run in all competitions to 38 games and seals their place in Friday's quarter-final draw for the ninth consecutive season.
Yet, they made a bright start with Mesut Ozil curling just wide of the target.
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Barca's front three soon came to life, though, and David Ospina made a brilliant save from Messi after the Argentine had perfectly controlled Neymar's floated pass over the Arsenal defence.
Jeremy Mathieu headed over a gilt-edged chance from the resulting corner.
Neymar wasn't so forgiving moments later when he was played in by Suarez and coolly slotted past Ospina for his 26th goal of the season.
Yet, Arsenal should have been at least level by the break as Barca struggled to handle the visitors pace on the break without the suspended Gerard Pique.
Alexis Sanchez missed a glorious chance to net on his return to the Camp Nou when he glanced a header wide from Hector Bellerin's cross.
Arsenal's pressure was finally rewarded six minutes into the second-half when Elneny rifled his first goal for the club into the top corner from Sanchez's layoff.
Conceding a goal sparked Barca back into life as Ospina got down well to his left to parry Messi's low drive.
Barca's nerves were settled in spectacular fashion by Suarez as the Uruguayan contorted his body to acrobatically volley home Dani Alves's cross.
Arsenal continued to create chances as Welbeck drilled a shot against the bar from a narrow angle.
Ter Stegen then produced a brilliant double save to turn away Sanchez's free-kick and Olivier Giroud's follow-up effort.
However, Messi again demonstrated the difference in quality between the sides in front of goal when he dinked home two minutes from time to take he, Neymar and Suarez's combined tally for the season to 106.