Mark Wright's goal kept Shrewsbury in the hunt for automatic promotion to League One.
The goal came on 10 minutes when Nicky Wroe ran 30 yards to the edge of the area before finding Wright, who calmly slotted home.
Cheltenham's Wes Thomas grazed the visitors' crossbar in the second half, but that was the closest the home side came to an equaliser.
Barnet's loss to Accrington confirms Cheltenham's League One safety.
Cheltenham Town boss Mark Yates told BBC Radio Gloucestershire:
"Both teams found the pitch difficult but the fans want to see effort and commitment to the cause and I think we gave that today.
"We conceded a poor goal on our part, it was finished well, but how we got to that position was not good.
"We should have had a penalty in the first half, we hit the bar and had a couple chances which sums up our season at home this year."