This result means three wins on the spin for Harps and they are now up to seventh in the table and 15 points ahead of bottom club Longford Town.
The opening quarter was a dull affair, with Galway’s only chance being a Padraic Cunningham shot well over the target while at the other end Keith Cowan’s thumping drive was blocked bravely by Stephan Folan.
Harps picked things up as the interval approached and Dave Scully had Conor Winn scrambling across the goal to save on 38 minutes.
The home side could have taken the lead on 59 minutes when the ball fell nicely for Scully but his shot was watched all the way by Winn who made a routine save, and shortly afterwards the former Harps custodian was called into action again to thwart Damien McNulty.
Harps broke the deadlock on 69 minutes when Josh Mailey slipped in a cross from the left for Sean Houston to latch onto with some conviction and fire past Winn.
Galway almost drew level three minutes later when Colm Horgan let fly with a cracking effort that had Brush fully extended to make an important save.
Brush; McNulty, Cowan, McMonagle, Mailey; Hanlon, Molloy, Houston, McNamee (Foy, 79 mins); Scully (McHugh, 87 mins), Curran (Keating, 68 mins).
Winn; Cantwell (Shanahan, 75 mins), Sinnott, Folan, Walsh; Horgan, Garcia, Connolly, Devaney; Faherty, Cunningham (Curran, 73 mins).
R Rogers (Dublin)