The champions-elect were forced to be patient against one of the league's surprise packages this season, but two goals in six second-half minutes extended their lead at the top of the table to 12 points.
La Blaugrana could be crowned champions as early as Wednesday should Atletico Madrid lose at home to Valencia, but if that doesn't happen then victory over Levante on Saturday would complete the job.
Ernesto Valverde chose to rest Lionel Messi ahead of next week's Champions League semi-final first leg against Liverpool, and Barca missed the Argentine's individual genius as they struggled to break a stubborn Alaves defence down for much of the first half.
The visitors did start brightly, though, and should have taken the lead inside six minutes when Philippe Coutinho pounced on a loose pass to release Suarez, who saw his left-footed strike saved by Fernando Pacheco before Coutinho's follow-up was blocked.
Barcelona got in behind again two minutes later as the same pair combined, but this time it was Suarez who saw his effort blocked by Rodrigo Ely after latching on to Coutinho's cut-back.
Alaves completed just five passes to Barcelona's 81 in the opening 10 minutes, but they began to stem the flow of chances as the half wore on despite Valverde's side continuing to see the lion's share of possession.
It was still Barcelona who looked most like scoring, though, and Coutinho drew a routine stop from range midway through the half before Sergi Roberto floated an effort wide of the far post when caught in two minds about whether to shoot or cross the ball.
Alaves fired a rare warning shot towards the end of the first half when they created their first sight of goal from a free kick, with Ely climbing highest inside the area but sending his header over the bar.
Any Alaves goal would have been completely against the run of play, though, and Barca did finally break the deadlock nine minutes into the second half when Suarez dummied Roberto's pass to allow it to run through to Alena, who took a touch before tucking his finish into the bottom corner.
A second goal arrived just six minutes later, but not before VAR played its part as Gerard Pique saw his initial goal ruled out for offside against Samuel Umtiti, with both defenders hooking the ball home from point-blank range.
The video replay proved the linesman right in his offside call, but also exposed a handball against Tomas Pina. The referee duly disallowed the goal but pointed to the spot, and Suarez made no mistake with his penalty to double Barcelona's advantage.
As if conceding twice in six minutes was not punishment enough for the hosts, Messi came on immediately after the second goal and was on the field for just eight minutes before hitting the woodwork as Pacheco got a crucial touch to turn his powerful strike against the post.
Alaves had a couple of chances to pull a goal back in the final 20 minutes as first Darko Brasanac fired over and then Jony curled an effort narrowly wide from outside the area, but it was a largely comfortable evening for Barcelona, who barely had to get out of second gear.
Indeed, the visitors perhaps should have made things more clear-cut when Messi and Suarez combined to tee up Vidal at the back post, only for the Chilean to send his header over the top.
Coutinho and Messi also missed the target when they should have done better in stoppage time, but by that stage the points were already in the bag as Barcelona remain on course to clinch the title before their Champions League showdown with Liverpool.
ALAVES (5-4-1): Pacheco; Vigaray, Laguardia, Ely (Twumasi 71'), Navarro, Duarte; Rolan (Brasanac 46'), Wakaso, Pina, Jony (Guidetti 81'); Baston
BARCELONA (4-3-3): Ter Stegen; Semedo (Alba 70'), Pique, Umtiti, Roberto; Vidal, Busquets, Alena (Arthur 87'); Dembele (Messi 61'), Suarez, Coutinho