Until Jordi Alba added the visitors' second late on, the game was very much in the balance, a fact not lost on Valverde.
"We had to win," he said to reporters after the game.
"Eibar pushed us very hard and were at a very high level, and I attribute our suffering to that.
"They were on an incredible streak of their own so the win has double merit."
With a Champions League match against Chelsea just around the corner, Valverde noted that his main objective was to get back to winning ways in LaLiga.
"This match was very important for us after two draws in the league," he continued.
"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."