Defender Luke Ayling was sent off in the 84th minute for handling Lawrie Wilson's cross, but Stech saved the resultant spot-kick from John Mousinho.
Chances were at a premium during a largely disappointing first half. The Glovers opened brightly with Gavin Massey firing narrowly wide and Kieran Agard going close, but Stevenage soon gained a foothold in the game. They kept Stech busy during the the remainder of the half.
Craig Reid was twice denied by the keeper, while Byron Harrison almost broke the deadlock as he fired inches wide shortly before the interval.
Yeovil opened the second period brightly and Paul Wotton went agonisingly close as his shot from distance only just flew wide.
At the other end, the hosts responded with Harrison finding space in the box before his shot missed the target by a couple of feet. Stech was to be tested by Mousinho again before his late penalty save from the same player.