East Fife went close in the opening minutes when a shot on the turn from Craig Smart was saved by Andy Reid.
In the fifth minute Paul Napier crossed for Andy Rodgers whose header was saved by John Dodds.
East Fife should have scored in the seventh minute when Stephen Fortune was clean though but shot wide of the post.
Just a minute later Paul Ritchie broke free on the right and fired a shot against the bar. Montrose went close in the 13th minute when a shot from Scott Michie was fumbled around the post by Dodds.
In the 27th minute Ritchie headed narrowly wide from a Ryan Blackadder cross and eight minutes later Michie controlled a Jered Stirling free-kick only to see his shot saved.
Montrose took the lead in the 37th minute when Rodgers crossed for Michie to score.
Montrose could have stretched their lead in the 53rd minute when Michie lobbed just over the crossbar.
In the 69th minute Blackadder had a shot saved by keeper Reid and five minutes later Neil Jablonski's header just cleared the Montrose crossbar.
Chances continued to be created at both ends and Rodgers had a shot pushed out by Dodds.
In the final minute Steven Kerrigan headed inches wide from a Napier corner and in stoppage time a Jonathan Smart header forced a super save from Reid.