The miss allowed the visitors to take the lead just three minutes later as Helder Postiga crossed the ball in for Dominquez to smash into the roof from close range.
Eibar kept pressing as half time approached, but only Raul Albentosa threatened with an equaliser, but his 35-yard rocket flew well high.
Dani Nieto came close to getting the hosts back on level terms just a couple of minutes after the restart when he won a corner, but he fired wide once the ball was crossed back in.
Eibar continued to look the more likely to score throughout the majority of the second half, but Abraham had a shot blocked as the home side failed to level the scores before full time.
The result gives Deportivo their first win since returning to the Spanish top flight, while Eibar, who are also newly promoted, sit 14th in the table.