Galway captain Fiontan O Curraoin hailed his side's belief after they edged past Roscommon at Dr Hyde Park.
Galway capitalised on Roscommon's failure to score from a host of presentable opportunities in normal time to rally with three late points from Shane Walsh to force extra-time. All the momentum was with the Tribesmen in the additional period and they duly wrested back the provincial crown.
"We just kept believing. We knew that if it came down to the end, we'd be able to fight and we did. Luckily, we grabbed it at the end," said a delighted O Curraoin afterwards.
Both teams made an explosive start. A shot from Cathal Mulryan inside the first minute was carried over the line by Roscommon goalkeeper Tadhg Lowe. But Roscommon responded instantly. Conor Daly's pass found Donie Smith and he finished to the net.
Roscommon found the net once more in the 19th minute. Smith did brilliantly to win back possession and his crossfield pass found Finbar Cregg, who steadied himself before lashing the ball home.
Roscommon led by 2-4 to 1-6 at the break and looked the more likely side to win following Eoin Walsh's dismissal, but they spurned a host of golden opportunities in the second half.
Donie Smith followed Walsh to the line for a second yellow card and Galway seized their lifeline with Shane Walsh stepping up to the plate late on. His equaliser was a spectacular effort after being set up by man of the match Ian Burke.
The visitors gradually pulled away from their opponents in extra-time as Roscommon wilted. By the midway mark, they were 1-14 to 2-9 ahead and when Burke, Shane Maughan and Walsh pointed in quick succession in the second period, there was to be no way back for Roscommon.
"After the sending off, we had Conor Daly as a free man and everything seemed to be going according to plan. The chances we missed, what can you do? They were unforgivable. Then, when Donie (Smith) was sent-off, we were under serious pressure," lamented Roscommon boss Nigel Dineen.
Galway now face Kildare in the All-Ireland semi-final.
SCORERS - Galway: S Walsh 0-7 (5fs), I Burke 0-6, C Mulryan 1-0, S Moran, P Varley, G Kelly, S Maughan 0-1 each. Roscommon: D Smith 1-2 (2fs), C Murtagh 0-4 (2fs), F Cregg 1-0, C Compton 0-2, E Smith, T Corcoran (f), M Holland 0-1 each.
GALWAY – T Healy; L Silke, J Shaughnessy, E Walsh; M Loughnane, G Canavan, P Varley; F Ó Curraoin, D Comer; C Mulryan, S Moran, S Maughan; S Walsh, A Varley, I Burke. Subs: T Flynn for Silke (27), C Rabbitte for Maughan (44), P Cunningham for Moran (45), P Ezergailis for Canavan (48), G Kelly for Comer (58), D Cunnane for E Walsh (f-t), D Comer for Mulryan (73), S Maughan for A Varley (75), M Heskin for Cunningham (80).
ROSCOMMON – T Lowe; D Murray, R Timothy, C Cafferky; M Connaughton, J McManus, C Daly; F Kelly, T Corcoran; F Cregg, C Connolly, E Smith; D Smith, C Compton, C Murtagh. Subs: S Killoran for Cregg (52), C Kilcline for E Smith (60), M Nally for D Smith (f-t), M Holland for Kelly (f-t), P Corbett for Killoran (65), C Cox for Connaughton (65), E Smith for Connolly (72).
REF - E O'Grady (Leitrim).