Danny Cummins was the hero for Galway as his goal 12 minutes from time fired Galway to a second FBD League win of the year on a scoreline of 1-08 to 1-05.
Despite their second win in a week, Galway still need a win in their final pool game against GMIT next weekend to reach the final.
A first-half goal from Nevin O'Donnell meant Leitrim only trailed by a point at the break, and three in-a-row gave them a lead by the 40th minute.
But Cummins' goal and point after his introduction midway through the second-half gave Galway victory at Tuam Stadium.
Meanwhile, Joseph Donnellan bagged the goal for GMIT as they shocked Sligo 1-10 to 0-12 at Collooney to keep their FBD League hopes alive.
Niall Murphy was brilliant in midfield for Sligo and scored six points for the second week running, but it wasn’t enough.
It was tied 0-6 a piece at half-time with midfielder Murphy keeping Sligo in touch. He kicked four points in the first-half while Joe Rodgers and Brendan Egan were also on target.
Galway's Martin Farragher was among the scorers for the defiant students and they converted their chances in the second-half, while Peter Cooke was the sole contributor in the first-half for GMIT.
Mayo and Roscommon will do battle next weekend for a spot in the final after both sides picked up their second wins of the campaign.
Stephen Rochford’s reign as new Mayo manager continued on a positive note as his side beat IT Sligo 1-16 to 0-8 in Ballina.
Despite Shane Nally’s black card in the second-half, Kevin McLoughlin’s goal sealed the win for the second week in a row.
Mayo held a comfortable 0-7 to 0-3 lead at the break after a dominant display in the first-half.
Elsewhere, Niall Kilroy didn’t kick Roscommon’s first score from play in the second-half until 19 minutes in but they held on for a 0-12 to 0-8 win over NUI Galway at Strokestown.
Mayo youngster Conor O’Shea starred at midfield for NUI Galway, however Roscommon’s experience stood to them.
Roscommon led 0-6 to 0-2 at half-time with Cian Connolly kicking three points after Niall Kilroy opened their account, while Ciaran Murtagh and Conor Devaney also scored.