The visitors - who needed to win by three points to make the last eight - looked set to make the knockout stages when they led 0-16 to 0-13 with five minutes remaining.
But a John McGrath free and an effort from Kevin O’Brien meant the Galway men’s bid feel agonisingly short and UL now advance to the last eight.
The highlight of the game was the battle of the marksmen, McGrath and Bobby Duggan, who each hit nine points.
UL manager Brian Lohan had mixed emotions at the final whistle: “We are through but it was a disappointing result for us. We did enough but I’m very disappointed with a few of the decisions made out there.
“But full credit to our lads, they worked hard and got themselves back into the game, with a lot of things going against us on the day.
“We have such a good bunch of players, a very committed group and they did well.”
NUIG were 0-13 to 0-6 clear at half-time after they dominated the first-half backed by a strong wind.
They were 0-4 to 0-1 up after eight minutes before UL got it back to the minimum momentarily, but four late frees from Duggan ensured NUIG were well in control at the break. When Duggan struck the first point of the second-half, a shock appeared on the cards
. But UL were not out of it yet, and with McGrath on form from placed-balls they brought it back to 0-15 to 0-13 after 48 minutes. Duggan’s ninth of the day looked to have secured the three-point win that NUIG required but crucially they failed to add to their tally.
And with McGrath and O’Brien on song, UL squeezed through.
“The lads showed fierce character out there. I’m very proud of them because they gave it everything. It was a brilliant performance by them, and it’s a pity when it finishes like this – win a game and be out,” said NUIG boss Tony Ward.
B Duggan 0-9 (0-8f); N O’Brien 0-2, P Mannion 0-2; C Mannion 0-1, K McHugo 0-1, O Donnellan 0-1.
J McGrath 0-9 (0-7f); T Morrissey 0-2; J Forde 0-1 (0-1f), K Hehir 0-1, K O’Brien 0-1, D Fitzgerald 0-1.
C Tuohy (Galway); C Ryan (Tipperary), B Fitzpatrick (Clare), R Lane (Galway); S Cooney (Galway), J Hanbury (Galway), P Mannion (Galway); N O’Brien (Westmeath), O Donnellan (Clare); B Duggan (Clare), K McHugo (Galway), M Mullins (Galway); J Fox (Limerick), C Whelan (Galway), C Mannion (Galway).
G Hennelly (Galway) for Fox (49 mins), S Hynes (Galway) for Lane (54).
P Maher (Tipperary); M Casey (Limerick), J Browne (Clare), S Roche (Galway); A McGuane (Clare), B Heffernan (Tipperary), B Stapleton (Tipperary); K Hehir (Clare), D Fitzgerald (Clare); J Forde (Tipperary), T Morrissey (Limerick), C Martin (Kilkenny); J McGrath (Tipperary), K O’Brien (Limerick), T Heffernan (Tipperary).
P Ryan (Limerick) for McGuane (38 mins), C McInerney (Clare) for Heffernan (42), B Maher (Tipperary) for Hehir (51).
Cathal McAllister (Cork).