Sean Kelly and captain Damien Comer struck the crucial injury-time scores for a gallant Galway side that held firm in tense circumstances to rack up back to back wins in the Super 8s.
Just a point separated the sides entering injury-time as Kildare, who lost Daniel Flynn to a 46th minute red card, dug deep and refused to give in.
But points from Kelly and Comer sealed Galway's second win in a week having dismissed Kerry at Croke Park in Phase 1.
It's the perfect start to the Super 8s for the Tribesmen who had 13 different scores on the afternoon.
But Kildare's hopes of reaching the last four of the Championship are over after losses to Monaghan and now Galway having famously ignited their summer here weeks back with a qualifier win over Mayo.
They sought a first Championship win over Kildare since 1926 but just couldn't pull it off as the league finalists' superior quality and big game experience ultimately won out.
Galway were rocked early on by the loss of former U-21 star Michael Daly to an apparent leg injury.
The wing-forward fired two points in the first eight minutes but pulled up in the process of scoring his second and appeared to tweak a muscle.
That scored left Galway 0-3 to 0-1 ahead though Kildare fought back to level terms at 0-3 apiece, setting in train a tit for tat scoring encounter.
It was thrilling stuff and the scores were tied on five occasions in a blood and thunder first-half that amounted to an exhibition of Gaelic football.
All of the best characteristics of the game; high fielding, long range point kicking, ultra intensity and quality kick passing were on show as the game swung from end to end.
Comer and Shane Walsh were marked men in the Galway attack though both still contributed 0-05 in the opening half, 0-03 from play.
Johnny Heaney lobbed over a beauty too and the Tribesmen closed out the first-half scoring with three in a row to lead 0-11 to 0-10.
But Kildare led on a number of occasions themselves and played their part in an epic opening half.
Daniel Flynn and Paul Cribbin kicked some stunning long range scores though six wides in the half cost them.
Galway opened up a three-point lead in the third quarter for the first time and were boosted by Kildare's loss of Flynn to the red card.
The former Aussie Rules man was picked out following an off the ball incident involving Galway full-back Sean Andy O Ceallaigh.
Galway remained in the ascendancy but a mixture of poor kicking and handling errors allowed Kildare to stay in touch with points from Paddy Brophy and Paul Cribbin.
Galway led 0-15 to 0-12 approaching the hour mark and moved four clear when Gary O'Donnell pointed but simply couldn't shake off a determined Kildare side.
Points from Brophy and Neil Flynn got Kildare back to within a point as the game moved into injury-time but Galway ultimately held on for a hugely significant win.
Galway scorers: S. Walsh (0-4, 3 frees), D. Comer (0-3), M. Daly (0-2), J. Heaney (0-1), C. Sweeney (0-1), P. Cooke (0-1), T. Flynn (0-1), G. Bradshaw (0-1), E. Brannigan (0-1), I. Burke (0-1), G. O'Donnell (0-1), S. Armstrong (0-1), S. Kelly (0-1).
Kildare scorers: N. Flynn (0-5, 3 frees), P. Cribbin (0-3), F. Conway (0-2), D. Flynn (0-2), P. Brophy (0-2), K. Flynn (0-1), C. Healy (0-1).
Galway: R. Lavelle; D. Kyne, S.A. O Ceallaigh, E. Kerin; G. Bradshaw, J. Heaney, C. Sweeney; P. Cooke, T. Flynn; S. Kelly, S. Walsh, M. Daly; I. Burke, D. Comer, E. Brannigan.
Subs: P. Sweeney for Daly (10); S. Armstrong for P. Sweeney (51); G. O'Donnell for Heaney (54); A. Varley for Brannigan (black card) (59); J. Duane for O Ceallaigh (67); F O Curraoin for Flynn (70).
Kildare: M. Donnellan; P. Kelly, M. O'Grady, D. Hyland; E. Doyle, K. Flynn, J. Byrne; K. Feely, T. Moolick; K. Cribbin, P. Cribbin, F. Conway; N. Flynn, D. Flynn, P. Brophy.
Subs: N. Kelly for Moolick (50); C. Healy for K. Cribbin (53); J. Murray for P. Kelly (59); D. Slattery for Cribbin (62), E. Callaghan for Brophy (black card) (68), F. Dowling for Conway (75).
Referee: S. Hurson (Tyrone).