Suarez opened the scoring against the run of play 16 minutes in, before Alba finally made certain of the points at the death after the in-form hosts went down to 10 men.
Three days ahead of their first-leg trip to Chelsea in the last 16 of the Champions League, Valverde was happy to see Barca pass the tricky test.
"We had to win. Eibar pushed us very hard and were at a very high level, and I attribute our suffering to that," he told reporters after the 2-0 victory. "They were on an incredible streak of their own so the win has double merit.
"This match was very important for us after two draws in the league. I don't care much about Chelsea, our objective was to win again. In Ipurua everybody suffers, nobody wins easy, but Tuesday is another story."
Barca, still unbeaten after 24 games, are now 10 points ahead of second-placed Atletico Madrid.