Sexton, who shared the Irish fly-half role with Munster veteran Ronan O'Gara during the RBS 6 Nations, missed an early penalty attempt before Lifeimi Mafi set up former All Black Doug Howlett for the game's opening try.
O'Gara added the extras before Sexton knocked over his first penalty as Munster led 7-3 at the interval.
Leinster looked to be taking control in the second half when Donncha O'Callaghan was sent to the sin bin for not rolling away and Sexton added two more penalties to put the European champions in front for the first time.
But with a quarter of an hour remaining, Keith Earls collected O'Gara's pinpoint cross-field kick before slipping past Isa Nacewa and Shane Horgan to put Munster back in front.
O'Gara missed the conversion and a penalty opportunity but the win was virtually sealed when Munster were awarded a penalty try for a collapsed Leinster scrum.
The try was duly converted by O'Gara and despite late attempts from Kevin McLaughlin and Nathan Hines, both denied by the video referee, Munster held on to join the Ospreys as three-time Magners League winners.