Leitrim fought back against a powerful Galway side but eventually ran out of steam going down by three points 1-11 to 1-08.
Michael Donnellan was impressive, kicking Galway into a four point lead midway through the first half.
Leitrim missed several chances to stay in touch with the best chance falling to Declan Maxwell who was unlucky as he thumped the ball wide of the Galway post.
A Donal Brennan goal in the 46th minute gave Leitrim renewed hope but Galway hit back with a Paul Clancy goal to widen the gap.
Leitrim brought the difference back from six to three points before time eventually ran out as Galway progressed.