Halifax turned in one of their best performances of the season and battered the visitors in the opening 45 minutes.
But all they had to show for it was a single goal from centre-back Greg Young, when the game should have been over at the break.
Stamp showed why Halifax boss Chris Wilder would love to have him in his side after he popped up five minutes from time to steal a point.
All that stood in the way of the Shaymen and a big half-time lead was keeper Alan Julian, who pulled off a string of top-class saves.
Halifax, unbeaten at home all season, were simply outstanding and Julian kept Stevenage in the game almost single-handedly.
The hosts had their own star man in on-loan winger Danny Forrest, who ripped into Stevenage only to be thwarted repeatedly by Julian.
Brian Smikle and Lewis Killeen were also denied by the keeper, but Halifax finally got their reward 10 minutes before half-time, though there was a stroke of fortune about the goal.
Young claimed his first of the season from a corner, but his header seemed to lack power. However, the ball bobbled through a crowd of players leaving Julian unsighted and sneaked in at the far post.
Defender Barry Laker was stretchered off in first-half stoppage-time with a hamstring injury and manager Graham Westley switched to three at the back.
His re-organisation changed the game and Stevenage edged the second half. With time running out Westley threw more men up front and Jon Nurse twice went close.
Just when it seemed Town would hold out Stamp rose to head home Dino Maamria's corner to keep Stevenage on course for the play-offs, while Halifax must wait a little longer to seal a top-five finish.