Ashley Nadesan's 25th minute strike separated the sides for most of the afternoon and looked set to be enough to win the game, despite the forward missing a 74th minute penalty.
Glenn Morris made an outstanding injury-time save to deny Lewis Alessandra and keep the Reds ahead but the hosts were not kept out for long.
Alessandra was the man involved again as he headed in from Ben Dickenson's 93rd minute corner.
It was a cruel blow to Crawley, who were so close to claiming a vital three points away from home, which would have been their fourth consecutive victory.
However, they remain unbeaten in 2022 and sit in 11th place, just six points behind the play-off places — with a game in hand.