The three points saw the Shrews widen the gap between themselves and the League One relegation zone to eight points, with other results involving teams below them also going their way.
James Dunne's scorching volley put Stevenage in front but former Wales international Paul Parry had Shrewsbury level by the break before Marvin Morgan, on as a late substitute, converted a cross from Jon Taylor to seal victory for Turner's side five minutes from time.
Shrewsbury's triumph, the first time they have won successive games this season, was confirmed by a brilliant stoppage-time save from goalkeeper Chris Weale to deny Mark Roberts an equaliser.
"It's a bit brighter looking at the table but it was hard work," said Turner. "We lived dangerously late on and there was one fantastic save from Chris Weale which earned us the three points.
"The victory lifts the pressure enormously from us and I think we can go into Tuesday's game against Doncaster, who are a good side and right up there, thinking that we can get another good result.
"You would expect one or two results elsewhere to go against us and Oldham won, but Tranmere winning at Colchester has done us a favour and there were one or two others. Hartlepool's result at Scunthorpe was a great one for us so it's just opened up a little bit of a gap now and given us a little bit of breathing space."