Two first-half tries by St Mary's inflicted enough damage on Wesley to win this tie at Lakelands Park.
Both tries came after good work by centre Jack Finnigan, who after five minutes made an impressive run down the left before finding his outhalf, Jonathan Sexton, who made it over the line.
Ten minutes later it was 10-0 after a similar move by Finnigan left full back Robert Gannon free to score.
The lead was extended when Sexton landed a drop goal eight minutes into the second half. Wesley's consolation points came near full time through a penalty by scrumhalf Ian Runtree.
St Mary's College: R Gannon; D Shaffrey, J Finnigan, T Daly, R Bowe; J Sexton, B McDermott; B McGovern, G O'Meara, D Fallon (capt), C Burke, B O'Kelly, H Queenan, J Kelly, C Lucas. Replacement: A Culligan for Queenan (58 mins).
Wesley College: P Galvin; R Millaney, R Meredith, S Odidison, S Macken; R Greene (capt), I Runtree; S Wall, K O'Byrne, W Corrigan, J Mokoo, R England, P Walker, P Marshall, M Maguire. Replacement: A Culhane for J Mokoo (42 mins).
Referee: P Fagan (Leinster).