The Glovers saw their seven-game winning home run come to an end after goals from Joe Walsh and Matt Sparrow cancelled out Byron Webster and Paddy Madden's first-half strikes.
The hosts took the lead after only five minutes when towering defender Webster powerfully headed home debutant Ben Gordon's cross.
Crawley's misery was compounded when Madden clinically converted Luke Ayling's delivery to notch his 20th goal of the campaign.
However, the visitors - who came into the game on the back of four wins in their last seven away games - threatened and missed some gilt-edged chances before half-time.
Billy Clarke rattled the crossbar with a free-kick before Brentford loanee Paul Hayes twice spurned glorious openings, including one from only five yards out.
Seconds after the break, though, the visitors reduced the deficit after Walsh nodded home a Mat Sadler free-kick.
Sparrow then completed the comeback with a confident strike before both sides rattled the woodwork twice in a dramatic late flurry.