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.
Alena and Suarez struck within the space of six second-half minutes to boost Barca ahead of their Champions League semi-final clash with Liverpool next week.
The Catalan giants will be confirmed as LaLiga champions for the eighth time in 11 seasons should nearest rivals Atletico Madrid lose at home to Valencia on Wednesday.
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.
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.
