A penalty awarded for a foul on Ade Akinbiyi by David Worrell and converted by substitute Jean Louis Valois five minutes from time gave Burnley the edge after Plymouth had looked the more likely scorers.
But with the visitors committed to attack late on it was left to midfielder James O'Connor to delight the home fans by drilling home a superb solo goal in injury time.
Earlier only the heroics of Brian Jensen had kept Burnley in the game. The big keeper first pushed a shot from Nick Chadwick onto the post and then pulled off an acrobatic effort to keep out a header from Plymouth substitute Steve Milne.
David Norris and Micky Evans also went close for the visitors, but the two late goals condemned Argyle to their 15th away defeat.