Tipperary star Jason Forde pointed 10 frees as UL booked a home quarter-final with ease.
Beaten finalists last season, Mary I trailed by 11 points at half time, 1-13 to 0-5, the goal scored by Clare hero Podge Collins midway through the half following a goalmouth scramble.
David Reidy and three pointed frees by Jamie Wall, who scored 0-12 overall, gave Mary I hope and, after Forde pulled one one back for UL – managed by former Clare star Brian Lohan – Wall scored three more frees to cut the gap to five points.
That was as good as it got, though, for Eamon Cregan's charges as Tommy Heffernan cracked in a second UL goal to clinch the win after 40 minutes.
Scorers – UL: J Forde 0-10fs, P Collins, T Heffernan 1-1 each, C Ryan 0-2 (1f, 1 '65'), P Walsh 0-4, J Glynn 0-2, B Maher, C Malone, B Stapleton, 0-1 each. Mary I: J Wall 0-12 (8fs, 1 '65'), C Galvin, D Reidy 0-2 each.
UL – S Hassett; P Flanagan, J Sheehan, J Browne; T Ryan, D Morrissey, C Ryan; B Maher, P Walsh; C Malone, P Collins, B Stapleton; J Forde, J Glynn, T Heffernan. Subs: S Moey for Stapleton, M Carmody for Browne, T O'Brien for Maher, M Ryan for Collins, P Geraghty for Heffernan.
Mary I – N Kelly; N Wynne, C Barry, A Ryan; L Corry, S Kennedy, j McGrath; C Galvin, T McGrath; D Reidy, J Hannon, L Considine; E Hogan, J Wall, D Ferncombe. Subs: B McCarthy for Hogan, C Treacy for Ferncombe, N Clery for McGrath, PJ Keane for Barry.
Ref – A Kelly (Galway)