The Pilgrims dominated the opening half but had to wait until the 43rd minute to take the lead courtesy of Reuben Reid.
However, Crawley pulled level five minutes from time when Ronnie Deacon bundled the ball home.
Argyle, though, claimed a deserved three points when on-loan Crystal Palace midfielder Hiram Boateng scored his first goal for the club in the 89th minute.
Manager Derek Adams said: “The first half was the best performance I’ve had this season. We were outstanding. We moved the ball quickly and created many opportunities in the game.”
Argyle are two points ahead of Wycombe and Leyton Orient in second and third place.
“We’re 11 games into the season and moving along nicely,” said Adams. “But it is tight at the top of the table.”