Micky Mellon’s men took the lead in the 20th minute when striker Jean-Louis Akpa Akpro’s low shot took a heavy deflection off Durrell Berry, giving Trevor Carson no chance.
Cheltenham missed a good chance seven minutes later when Wes Burns headed Matt Richards’ cross wide after a short corner taken by Matt Sparrow.
Shaun Harrad went down in the box under a challenge from Jermaine Grandison in the 38th minute, but a free-kick was given against him and Shrews goalkeeper Jayson Leutwiler denied Burns in first half stoppage time.
Harrad went close with a free-kick and Zack Kotwica saw a drive well blocked by Mark Ellis before Jayson Leutwiler kept out Harrad’s header with a fingertip stop.
Carson made an excellent save to deny Tyrone Barnett in the 85th minute before Shrewsbury’s celebrations started and the Robins fell out of the Football League after 16 years.