St Thomas had edged a point in front from Galway senior James Regan when Cooney burst through to blast into the top right corner.
But Gort fought back and, after Gerry Quinn fired a 20 metre free to the net in the second minute of stoppage time, they almost snatched it at the death when another Quinn free was deflected over the bar in a crowded goalmouth.
All-Star David Burke was superb for St Thomas and helped them race into a 0-4 to 0-1 lead after eight minutes. However, Quinn’s accuracy saw Gort come back and the sides were level 0-8 each at the interval.
Gort hit three points after the restart but St Thomas again fought back and a couple of frees from another Galway player, Conor Cooney, saw the sides level at 0-12 each going into the final quarter.
Cooney was set up by another minor, substitute Eanna Burke, and his wonder goal set up a county final meeting in two weeks against another of their neighbours, Loughrea.
St Thomas: P Skehill; C Burke, R Murray, S Burke; E Tannion, S Skehill, Darragh Burke; K Burke (0-2), David Burke (0-2); J Regan (0-2), C Cooney (0-5, 4fs), B Burke (0-1); G Murray, R Murray (0-3), A Kelly.
Subs: S Cooney (1-0) for G Murray, 29; E Burke (0-1) for Kelly, 39.
GORT: P Cummins; C Helebert, M McMahon, M Cummins; J Grealish (0-2), B Regan, S Óg Linnane; G Lally, A Harte; G Quinn (1-9, all frees), P Killilea (0-2), M Nestor (0-1); R Cummins (0-1), O Fahy, G O’Donoghue.
Subs: A Mullins for Fahy, 48; K Killilea for O’Donoghue, 49.
Referee: S Moran.