Canada led 19-16 with just 11 minutes remaining, but two late scores were enough for Ireland A to pull through.
Murphy, playing in his second Churchill Cup, was included on the wing by national coach Declan Kidney and played the full 80 minutes.
Ireland’s next fixture is against Georgia on Sunday.
Ireland A
15 Denis Hurley (Cork Constitution/Munster)
14 Fionn Carr (Galwegians/Connacht) rep Frank Murphy (Buccaneers/Connacht) 62min
13 Fergus McFadden (UCD/Leinster)
12 Keith Matthews (Buccaneers/Connacht) rep James Downey (Northampton Saints) 80min
11 Johne Murphy (Leicester Tigers)
10 Niall O'Connor (Belfast Harlequins/Ulster)rep Jonathan Sexton (St Mary's College/Leinster) 64min
9 Isaac Boss (Ballymena/Ulster, captain
1 Bryan Young (Ballymena/Ulster) rep Cian Healy (Clontarf/Leinster) 57 min
2 Sean Cronin (Buccaneers/Connacht) rep Denis Fogarty (Cork Constitution/Munster) 62min
3 Tony Buckley (Shannon/Munster)
4 Donnacha Ryan (Shannon/Munster) rep Trevor Hogan (Blackrock College/Leinster) 80min
5 Devin Toner (Lansdowne/Leinster)
6 Neil Best (Northampton Saints)
7 Sean O'Brien (Clontarf/Leinster) rep John Muldoon (Galwegians/Connacht) 72min
8 Chris Henry (Ballymena/Ulster)
