Jon Stead volleyed home for County in the first half, but his effort was ruled out for offside.
Fraser Franks produced Stevenage's best effort from Chris Whepldale's free-kick, but Scott Loach saved superbly.
With time running out, Liam Noble rifled into the corner from the edge of the area to give County the points.
The defeat left Stevenage 22nd in the table, but nine points clear of the relegation places with five games remaining, while County climbed to 18th.
Notts County secured their first home win of 2016, and first under new manager Mark Cooper, with a late goal against Stevenage.