Cooney hit the net in the first half to settle the champions after an unconvincing start and he finished strongly for a tally of 1-7 as John Burke’s side showed brief glimpses of the form that led them to a first Galway senior crown last year.
The victory was not secured until the closing moments when the youngest of the six Burke brothers, Eanna, poked to the net to crown an impressive performance.
St Thomas’ chances of making the quarter-final were hampered by results elsewhere as title favourites Portumna were defeated by 2010 All-Ireland champions Clarinbridge.
Paul Coen hit 1-11 for Clarinbridge, with the decisive score arriving on 52 minutes from a penalty after Galway senior Barry Daly was fouled.
Portumna had a super chance in the closing staged to seal a draw, but Galway veteran Ollie Canning saw his point blank effort sail wide, as Clarinbridge took a 1-14 to 0-14 victory, that sees them on top of A.
Ardrahan and Padraig Pearses became the first two sides to reach the quarter-finals after both sides picked up convincing victories.
Goals in each half from Gary Lally and a Fergal Flannery penalty gave Pearses a 2-11 to 0-14 victory over Beagh, confirming the the East Galway side will finish top of Group D.
In Group B, Ardrahan eventually clinched their spot in the last eight with a 2-9 to 2-8 win over Gort. Turloughmore will be favourites to join Ardrahan in the quarter-finals after a 2-18 to 0-17 win over Mullagh.
Debutant Sean Linnane pointed after barely 20 seconds, and Ronan Badger hit the net in the third minute. Mullagh, for whom Davy Glennon hit two first-half points. When Gary Burke had Turlough’s second goal five minutes from the break, the win was all but secured.
In Group C Craughwell must be satisfied with a run in the shield competition after they suffered a 1-18 to 2-12 defeat to Kinvara. Two goals in the first-half gave Stephen Glennon’s side a one-point lead at the break, but after Steve Moylan was sent off 15 minutes from time for Kinvara, the seasiders hit four points without reply late on to take the win.
Two goals from Enda Concannon was not enough for Castlegar as a late Neil Keary goal and a Johnny Maher free sealed a share of the spoils fro Loughrea. The sides now site level at the top of Group C, and both look poised to reach the knockout stages.