The victory takes Barcelona above Real Madrid into second place in the standings on 62 points, four behind leaders Atletico Madrid with 10 games left to play.
Lionel Messi has celebrated becoming Barcelona's highest appearance maker of all time by scoring twice as his rampant side crushed Real Sociedad 6-1 away in La Liga with a stunning team display.
Antoine Griezmann opened the scoring by netting against his old club in the 36th minute and right back Sergino Dest stretched Barca's lead shortly before halftime after latching on to a sweeping pass from Messi.
Dest struck again early in the second half before Messi controlled a sumptuous pass from Sergio Busquets and scored to mark a night in which he overtook Xavi Hernandez as Barca's record appearance maker, moving on to 768 games with the club.
Ousmane Dembele struck his side's fifth goal of the night in the 71st minute after a brilliant solo run while Sociedad's Ander Barrenetxea hit arguably the best goal of the game to give his side a rare moment of joy.
Messi, however, had the final say, finishing off a flowing team move in the 89th minute to score his 23rd league goal of the season, increasing his lead at the top of the scoring charts.
The victory took Barca above Real Madrid into second place in the standings on 62 points, four behind leaders Atletico Madrid with 10 games left to play.
Messi stays ahead of Suarez in the La Liga scoring charts, despite the Uruguayan's excellent finish nine minutes into the second half against Alaves.
Thomas Lemar turned the ball around the corner to Marcos Llorente, who fed Kieran Trippier racing down the right, and he curled a cross in on the run to the near post, where Suarez headed in.
But without a second goal, Alaves always had a chance and they were given a golden one after Stefan Savic was judged to have put an arm across Luis Rioja just inside the penalty area.
Joselu stepped up and struck it well to his left but Oblak read it right and, with his left foot still pinned to the line, sprung out and pushed the ball away.