The Uruguay international last found the back of the net in Europe in last term's incredible 6-1 victory over Paris Saint-Germain, but he restored Barca's three-goal cushion in the 87th minute after Edin Dzeko grabbed a late away goal.
Ernesto Valverde welcomed Lionel Messi, the match-winner against Chelsea in the previous stage, back into his starting XI following injury, but the Catalan side toiled until Roma shot themselves in the foot either side of half-time.
Messi was involved on both occasions as Daniele De Rossi and Kostas Manolas put through their own net, settling what had been a rather nervy Camp Nou.
Gerard Pique got the third before the hour mark and, following Dzeko's strike 10 minutes from time, Suarez hit the target in the Champions League for the first time in over a year.
Roma's early resolve was admirable in the absence of injured duo Radja Nainggolan and Cengiz Under, but they look to have too much to do back in Italy, with the chances of Eusebio Di Francesco's men mustering the firepower to down the LaLiga leaders extremely slim.
Barcelona made a slow, sloppy start and were initially restricted to shots from distance, Messi's 25-yard drive stinging Alisson's palms before an Ivan Rakitic curler clipped the base of the right-hand post.
Manolas claimed before the match there is "no way" to stop Messi, but the centre-back continued to effectively restrict Barca's main man, although he was able to tee up Suarez for an effort Alisson saved well.
It was typical then that the latest in a series of desperate defensive challenges would undo Roma's hard work seven minutes before the break, De Rossi lunging to keep the ball from Messi and inadvertently firing a rocket beyond his own goalkeeper and into the bottom-right corner.
The visitors should have drawn level seconds after the restart when Alessandro Florenzi's deep centre from the right reached the far post, but Diego Perotti directed a straightforward header back across the face of goal and wide.
More misfortune did for Roma in the 55th minute, though. Manolas dived in front of Samuel Umtiti in the area - after Messi combined with Rakitic - but clumsily bundled the ball against his own post and over the line.
The third goal followed four minutes later when Pique charged forward and tapped into the net when Alisson parried from Suarez.
Roma rallied late on and looked to have salvaged some interest in the tie, Dzeko holding off Alba to slide Perotti's pass into the bottom-left corner.
Suarez's timely contribution followed, though, with Roma unable to scramble clear and the striker prodded a finish under Alisson to put Barca within sight of a semi-final spot.
30 - After failing to score with his first 30 shots in the Champions League this season, Luis Suarez finally scored with his 31st shot. Ended.— OptaJoe (@OptaJoe) April 4, 2018
Key Opta stats:
- Barcelona have benefited from five own goals this season, a record in a single Champions League campaign.
- Roma are just the fourth side in Champions League history to score two own goals in a single game and the first since Galatasaray against Astana in September 2015.
- This was Andres Iniesta's 21st appearance in the Champions League quarter-finals, a joint high with Ryan Giggs and Xavi.
- Luis Suarez scored his first Champions League goal after a barren run of 1,058 minutes.