Munster will take on defending champions the Stormers from South Africa for the trophy and are the only team to beat the latter at home in the URC, having won 26-24 in the pool stages last month.
Leinster led 10-6 at halftime after lock Jason Jenkins crossed for the opening try from a barnstorming run, but Munster hit back after the break as another second row Tadhg Beirne finished some concerted pressure.
The visitors belied their status as underdogs as they dominated possession and created numerous entries into the Leinster 22, but were undone by some poor execution.
It looked as though they would be made to pay when Leinster lock Joe McCarthy scored a try just past the hour-mark but as Munster remained camped in their opponent's half, they worked a drop-goal position for Crowley and he put the ball between the posts and win the match in dramatic fashion.
Munster will be seeking their fourth title in the URC and first since 2010/11, while Leinster will have to get over their disappointment quickly as they face La Rochelle in the final of the European Rugby Champions Cup at their Dublin home on May 20.
Champions Stormers beat Connacht
Flyhalf Manie Libbok scored two tries as the defending champion Stormers held off a second-half fightback from Irish side Connacht to win 43-25 on Saturday and book a second successive final berth in the United Rugby Championship.
The Cape Town-based Stormers scored six tries in all as wing Angelo Davids, replacement scrumhalf Paul De Wet, loose-forward Marcel Theunissen and centre Ruhan Nel also breached the Connacht defence in front of 47,000 raucous home fans.
Connacht rallied from 20-8 down to pull within four points of the Stormers going into the final 15 minutes, scoring tries through wing Mack Hansen, loose-forwards Conor Oliver and Shamus Hurley-Langton and replacement back Byron Ralston, but they were unable to fully capitalise on multiple handling and set-piece errors by their hosts.
