Galway had the wind in their favour for the first-half in Salthill, but Peter Creedon's side brought plenty of bodies back behind the ball, resulting in The Tribesmen only holding a one point lead at half-time.
Alan Mullholland's men also did not help their cause by hitting nine wides in the opening 35 minutes, but they regrouped at the break and took control of the match in the second half.
Paul Fitzgerald had already denied Michael Meehan a certain goal with a brilliant save, but the decisive goal arrived in the early period of the second half.
Points from Mark Hehir and Sean Armstrong had pushed Galway 0-08 to 0-05 in front, when a free by Danny Cummins floated into the area.
Fitzgerald punched the ball, but it fell to debutant Michael Farragher and he stuck it low into the right of the net.
That was Galway's first goal in the qualifiers since 2007 v Meath.
After 52 minutes the home side were 1-11 to 0-06 in front and were threatening to pull away from Tipperary.
However, points from Conor Sweeney (who top scored for Tipp with 5, 2 from frees) and sub Bernard O' Brien ensured that Tipperary made it difficult for Galway all the way until the final whistle.
