Substitute Declan Edwards snatched a late winner, his first goal for Galway, leaving unlucky Drogheda five points adrift at the bottom.
The Boynesiders had an early scare when Vinny Faherty fired against the bar, but they dominated for long spells, despite having no recognised striker due to injuries.
Paul Crowley could have had a hat-trick, but was denied a number of times by Galway keeper Barry Ryan, who also foiled Paul Shiels in the second half.
Eleven minutes from time, Edwards scrambled in a header from a Derek O’Brien cross.
But the Drogs had a great chance to grab a point late on, but substitute Mark Salmon miskicked with the goal gaping.
