A try by John Madden in the final minute of the game for Munster saw Gearoid Lyons conversion attempt hit the post to deny the home side their first victory in this season's championship.
Captain Jack O'Donoghue scored Munster's other try and Lyons converted and also added two penalties.
Munster final game is away to Connacht next Friday in the Sportsground.
Munster: David Johnston (UL Bohemian); Alex Wootton (Exile), Ned Hodson (UCC), Dan Goggin (Young Munster), Adrian Enright (UL Bohemian); Gearoid Lyons (Shannon), Ryan Foley (Cork Constitution); Aidan Slattery (Cork Constitution), Ben Burns (UCC), Rory Burke (Cork Constitution); John Madigan (Dolphin), Darragh Moloney (Cork Constitution); Lee Nicholas (Shannon), Aidan Butler (Shannon), Jack O'Donoghue (Captain) (UL Bohemian).
Replacements: Max Abbot (Cork Constitution), Mikey Lynch (St. Senans/UL Bohemian), James Maxwell (Cashel), Patrick Kearns (Shannon), Sonny Dwyer (Cork Constitution), Aaron Atkinson (UCC), Sam Sterling (UL Bohemian), Greg O'Shea (Shannon).
Leinster: Billy Dardis (UCD); Cian Kelleher (Lansdowne), Adam Byrne (UCD), Garry Ringrose (Blackrock College), Ian Fitzpatrick (Lansdowne); Ross Byrne (UCD), Nick McCarthy (St. Michael's College); Ivan Soroka (Belvedere College), Dylan Donnellan (UCD), Oisín Heffernan (Lansdowne RFC); Ross Molony (UCD), Stephen Gardiner (Lansdowne); Josh Murphy (UCD), Michael Triggs (Lansdowne), Peader Timmins (UCD).
Replacements: Shane Delahunt (Lansdowne), Denis Coulson (Lansdowne), Craig Trenier (Terenure College), Stephen Keelan (Clongowes Wood College), Sean Coughlan (Blackrock College), Charlie Rock (Old Belvedere), Conor McKeon (Lansdowne), Stephen O'Neill (Terenure College).