Lionel Messi and Jordi Alba were among the stars named on the bench by Ernesto Valverde, who rested Ivan Rakitic entirely, but Barca eased to victory at Mendizorroza on Tuesday.
Suarez and Philippe Coutinho had early efforts blocked and the striker then saw his shot kept out by the legs of goalkeeper Fernando Pacheco after he was fed by Ousmane Dembele.
Arturo Vidal headed Dembele's right-wing cross over with Gerard Pique seemingly in a better position behind him as Barca sought the breakthrough.
Barca had created few chances in Messi's absence but they took the lead in the 54th minute.
Sergi Roberto, who switched from left-back to right-back at half-time, set up Alena with a clever pass and he beat Pacheco with a placed left-footed finish.
It was soon 2-0 in bizarre circumstances, Pique having a goal ruled out for offside as a VAR review instead led to a penalty for a handball by Tomas Pina.
Pacheco chose correctly in diving to his left but Suarez's penalty beat him for pace, and despite Barca taking a two-goal advantage Valverde sent Messi on for Dembele.
Messi and Vidal failed to add to the scoreline in the closing stages but the champagne can go on ice with the title virtually secured.
What does it mean? Barca close in on possible treble
It is fair to say Barca have not always been at their fluent best this season, but their consistency has to be admired. Since Valverde took charge in 2017 they have only lost three times in LaLiga.
Atletico and Real Madrid have failed to muster a genuine challenge this term, but Barca will not mind as they have been able to rest players including Messi for the Champions League.
Valverde has continually dismissed suggestions the treble is on but with Barca already into the Copa del Rey final they have a serious chance of repeating the feat achieved under Pep Guardiola and Luis Enrique.
Alena seizes his chance
One of the criticisms of Valverde has been his perceived failure to blood young talent, but Alena made his fifth LaLiga start in Vitoria.
The 21-year-old will surely get more minutes in the league with the title all but assured and four games still to play.
Pina the fall guy
Alaves competed but Pina conceding a penalty gave Barca the chance they needed to secure all three points. He may have thought he had got away with a handball before VAR intervened.
Key Opta Facts
- Suarez joined Ladislao Kubala in third on the list of leading LaLiga goalscorers for Barca (131 goals).
- Barca are the first a team to keep five consecutive away clean sheets against Alaves in LaLiga.
- Among LaLiga defenders, Only Alba (14) has more assists than his team-mate Roberto (seven) this season in all competitions.
- Alaves are without a victory in their last six LaLiga games (D2 L4), their worst winless streak under Abelardo Fernandez.
- Arturo Vidal won 16 duels against Alaves (25 in total), more than any other Barca player in a single game this season in all competitions.
Barca are at home to Levante on Saturday, the same day Alaves go to Athletic Bilbao, but Valverde's attentions may now turn to Liverpool as the treble becomes more realistic.