The victory lifted the home side out of the League One relegation zone and eased the pressure on head coach Sean O'Driscoll.
The former Arsenal player then turned goal-maker as Bryan beat the offside trap to race onto his pass and hammer an 87th minute shot past Jones with his left foot.
Jamie Proctor twice went close for Crawley between the goals, having a header tipped over and shooting over from a good position.
The first half was a bad advertisement for League One football as both sides gave the ball away continuously and barely mounted a worthwhile attack between them.
The only goal attempt of note came from Crawley's Emile Sinclair, whose powerful drive from the edge of the box was parried to safety by City goalkeeper Elliot Parish after 16 minutes.
- Sports & Recreation
- Kyle McFadzean