In a to-and-fro game, Luke Norris put Boro ahead before Reds captain George Francomb equalised soon after.
Norris added his second in the second half but Crawley levelled once again through Terence Vancooten's own goal.
Stevenage looked to have secured the win with nine minutes to go thanks to Arthur Read's goal.
But Jack Powell rifled home in the second minute of second half stoppage time to take a share of the points for Crawley.
This afternoon's results sees the Reds move up to 12th in the table with one game to go.
Manager John Yems made one change from last weekend's goalless draw at home to Forest Green Rovers. Nick Tsaroulla replaced Sam Matthews.
The hosts opened the scoring from the penalty spot on 21 minutes. Jordan Tunnicliffe brought down Elliot List in the box and Norris converted the penalty.
But the Reds levelled just three minutes later with a move straight from the training ground.
Powell's driven corner found skipper Francomb tucked home.
Stevenage restored their lead on 74 minutes. Vancooten found Norris who delicately chipped the ball over the head of keeper Glenn Morris.
Crawley hit back with another quickfire equaliser. Vancooten had the misfortune of putting the back into the back of his own net with ten minutes remaining.
The hosts struck what seemed to be the winner a minute later. List played in Read who curled a shot into the bottom corner.
But, in the second minute of time added on, Powell arrowed his effort from outside the box into the top right corner to snatch a point for the Reds
Crawley Town: Morris, Francomb, McNerney (Rodari 86), Tunnicliffe, Craig, Hessenthaler, J Wright, Powell, Tsaroulla (Davies 67), Maguire-Drew (Frost 67), Nadesan.
Unused: Nelson, Sesay, Allarakhia.