The Argyle skipper was in the right place at the right time to head a Barry Roche clearance into an empty net following the Morecambe goalkeeper's poor punch.
Luke Young gave the Pilgrims the lead with just four minutes on the clock. The young winger was allowed a free run on goal after Danny Carlton lost possession and he made no mistake from the edge of the box.
Midfielder Ladjie Soukouna missed an open goal after 22 minutes before the home side came back into the game in the second half. Jordan Burrow made it 1-1 after 51 minutes when he headed home an excellent right-wing cross from Izak Reid.
The goal sparked Morecambe into life and they took the lead with 19 minutes remaining when substitute Lewis Alessandra ran onto a long through-ball and chipped over the head of Ollie Chenoweth.
Morecambe looked set for only their seventh home win of the season until Purse was gifted his late equaliser.