The hosts took control with goals from Adams and Hope Akpan, but two goals in quick succession from Tom Hopper and Andy Bishop pulled bottom-of-the-table Bury level.
Adams, who began his career at Bury, gave his side victory with his fifth goal in four games, scoring with an attempted cross with the ball looping over the head of stranded keeper Trevor Carson.
Crawley broke the deadlock two minutes before the break when the unmarked Adams fired into the roof of the net from seven yards after being set up by a cross from Mike Jones.
The hosts doubled their lead eight minutes after half-time when midfielder Akpan latched onto a pass from Adams to rifle a low 15-yard shot into the bottom-left corner.
Bury dramatically pulled level with two goals inside three minutes - Hopper heading in a cross from Bishop on 66 minutes before Bishop (69) headed in a cross by substitute Craig Jones. However, the fightback was in vain as Adams sealed all three points for Crawley at the death.