David Ball handed Fleetwood the lead with just 90 seconds on the clock, the striker getting the final touch after an audacious overhead kick by Jamie McGuire.
The home side could have doubled their advantage 10 minutes in, a fine run by Tom Barkhuizen releasing Ball to have a deflected shot hacked off the line by Durrell Berry.
The visitors came into the contest as Paris Cowan-Hall proved a handful down the right, full-back Dean Howell twice left in his wake, and Scott Davies made vital saves to maintain Fleetwood's lead.
For all their possession Carl Fletcher's men failed to carve out clear chances and with less than a quarter of an hour remaining Purse picked up his second yellow in quick succession - a minute of madness seeing him floor Jamie McGuire and then tangled with opposite number Steve McNulty.
Fleetwood wasted no time in doubling their tally - Curtis Obeng breaking down the right for his low cross to be turned home by Plymouth's Curtis Nelson. Junior Brown popped up at the back post deep into injury time to add a third for Micky Mellon's men.