Peter Hutton scored his second goal of the season to move Shelbourne to within three points of leaders Shamrock Rovers at the top of the table.
Hutton struck for the only goal of a game Shelbourne dominated two minutes into the second half. Right back Hutton was initially involved as Shelbourne broke out of defence.
And the Derryman's pace got him down the field and into the box to receive Wes Houlihan's final pass before shooting past Mark Cobey.
Shelbourne ran the first half on a very wet and soft pitch but failed to find a way past an inspired Cobey who made several good stops.
Cobey first thwarted Shels as early as four minutes when Jonathan Minnock split the home defence to put Stephen Geoghegan through on goal. But Cobey was off his line quickly to save Geoghegan poor effort.
Having shown his alertness and pace, the Galway keeper then had to demonstrate his agility to keep out a stinging close-range shot from Pat Fenlon after Peter Hutton picked him out with a sublime cross from the right on 23 minutes. A further save from Cobey was required 10 minutes later from Tony McCarthy.
Shelbourne captain Owen Heary then twice came close to putting the gloss on Shelbourne's win. His shot from 20 yards on the hour dipped just over the crossbar. Then, 12 minutes from time, Cobey allowed Heary's well-struck 35-yarder squirm under his body and out for a corner.
Galway almost stole an equaliser in stoppage time when substitute Derek O'Brien went on a run into the Shelbourne box only to see Steve Williams make his only real save of the game to turn his shot out for a corner.
GALWAY UNITED: Cobey; Keady, Silke, Clery, Carberry (Quirke 62 mins.); Ryan, Herrick Carter (O'Brien 75), Murphy; Sheridan, O'Shaughnessy (Kane 68).
SHELBOURNE: Williams; Hutton, McCarthy, Doherty, Heary; Houlihan, Crawford, Gannon, Fenlon, Minnock; Geoghegan.
Referee: A Kelly (Cork).