Joaquin Sanchez came close for the visitors twice in the opening 20 minutes, first with a free kick that was well saved by Andres Varas and then striking the woodwork with another effort, the rebound of which was put wide by Isco Alarcon.
Celta could have been ahead on the half-hour mark when defender Javi Tunez was presented with an unmarked header inside the box, but he put his effort wide.
The hosts came within inches of taking the lead less than 10 minutes into the second half when Quinque de Lucas struck the woodwork after a quick break.
Malaga responded by coming just as close with 15 minutes left, this time Isco firing a powerful shot that came back off the upright.
The deadlock was finally broken seven minutes from time as Diego Buonanotte crossed the ball into the danger area and substitute Fabrice bundled the ball home, making him the youngest scorer in La Liga history.
Celta were desperately unlucky not to grab an equaliser in the last minute as Gustavo Cabral's header once again rebounded off the woodwork.