Cheltenham stay sixth in the table but they would have been looking to win this - their game in hand on the teams above them - to cement their position among the top seven.
Wrexham, meanwhile, will see this as a chance lost to make up ground on the leading sides.
The visitors drew first blood with a goal after only 56 seconds after Cheltenham had already hit the post.
Home defender Michael Townsend met a Grant McCann free-kick with a header that smacked off the woodwork.
Wrexham immediately got the ball downfield with Dean Bennett laying a pass through for on-loan striker Matt Derbyshire.
All the Cheltenham defenders appealed for offside but the flag stayed down and Derbyshire beat goalkeeper Shane Higgs with a low finish.
Cheltenham did most of the attacking for the remainder of the first half and were rewarded on 35 minutes when McCann rammed home a shot from 15 yards after Brian Wilson and Kayode Odejayi had combined to set him up.
Cheltenham got their noses in front 10 minutes after the break when Steve Guinan smashed in from close range after Wrexham goalkeeper Michael Ingham had spilled a Jeremy Gill cross.
The Robins should have taken a 3-1 lead when Odejayi found himself clean through with only Ingham to beat after some clever build up play but placed his shot wide.
The visitors made Cheltenham pay for that miss on 65 minutes when a lofted pass from substitute Darren Ferguson found the impressive Derbyshire and he beat Higgs with lob shot from the edge of the box.
After that, Higgs had to dive to deny Derbyshire his hat-trick but Guinan came mightily close to winning it with a header that bounced back off the post with 10 minutes left.