Pat Sullivan bagged the only goal of the game as Shamrock Rovers grabbed their first win of the campaign at Terryland Park.
Sullivan's seventh-minute header took a slight deflection, which left keeper Barry Ryan with no chance.
Under pressure to grab a result, Rovers started impressively, immediately taking control of the contest with their set pieces causing Galway problems.
In the fifth minute Sean O'Connor fizzed a free-kick into the box where Billy Dennehy made a powerful connection, but Galway keeper Barry Ryan saved.
The Hoops continued to pour forward, and they were rewarded two minutes later when they took the lead courtesy of a Sullivan header.
O'Connor again thumped a delightful free into a crowded goalmouth and Sullivan reacted smartly to put the Hoops ahead.
Galway responded and competed well for the remainder of the opening period as midfielder Stephen O'Donnell spearheaded the western challenge.
On the half hour mark, Ciaran Foley's hack forward was misjudged by Dan Murray and although Cian McBrien raced clear the Galway youngster failed to test Rovers keeper Alan Mannus.
McBrien had a 40th-minute snap shot saved by Mannus following good approach work from Seamus Conneely and Bobby Ryan, but the Hoops were a threat throughout.
Just before the break Robert Bayly blasted a long-range effort on target, which forced Ryan to make a save.
Four minutes after the restart, Dennehy went close again with another volley that crept inches wide.
O'Connor then examined Ryan with a blistering 64th-minute free kick.
In the closing stages Rovers were resolute and a vital away victory was posted.
