The visitors led through Adrian Mallagaray's strike but Ryan McStay drilled in Annan's leveller.
Bryan Gilfillan converted a spot-kick for the hosts and Matt Flynn's effort opened up a two-goal advantage.
Ross Campbell's penalty gave Montrose hope and, after Ben Jago was dismissed for fouling Campbell, Gary Fraser also scored from the spot.
Two late penalties gave Montrose a draw against Annan Athletic in Scottish League Two.