Following Saturday's 2-1 reverse at Exeter, Gary Megson's men lost again after a goal from Scott Laird just before half-time.
Wednesday's Chris O'Grady threatened to score in the second minute when he fired in a shot on the run which was tipped over by Chris Day. The Boro goalkeeper was also called into action to keep out Jermaine Johnson's firm drive.
Stevenage's Michael Bostwick put a couple of efforts off-target and had a shot saved by Stephen Bywater before the visitors scored the only goal of the game in the 45th minute.
Joel Byrom's corner was met by a far-post header from Chris Beardsley and Laird turned the ball past Bywater from close range.
David Prutton went close to scoring with a close-range shot which Day managed to keep out. Wednesday continued to put the visitors under pressure with O'Grady and Chris Lines both threatening but they were unable to level.