Zander Diamond ignited Northampton's play-off hopes against one of their rivals to send Town into the break in front.
But Stevenage replied through Whelpdale on 65 minutes and then, in a dramatic finale, secured all three points through Simon Walton.
Ryan Cresswell saw red in the aftermath and the Cobblers ended the day seven points behind the top seven with just four games remaining.
They travel to face Oxford United away on Tuesday evening.
Reaction: Chris Wilder named an attacking line-up for the game and admitted afterwards that it had been 'win or bust' for his side as they attempted to make the play-offs.
He told the official NTFC website: "We’ve been playing cup tie, win or bust football for quite a while now and we’ve done well to put ourselves in a position where we’ve been challenging for the play-offs but when we’ve not played well we’ve needed to grind out results."