In a game of few opportunities neither side could make the breakthrough needed to take the three points.
The visitors had the best of the chances and created the clearest opening of the game just after the restart when Shaun Harrad teed up Kaid Mohamed but the striker failed to find the target from six yards.
Morecambe's best effort came in the 52nd minute when Lewis Alessandra forced Scott Brown into an excellent save low to his left from a rare counter attack.
Cheltenham started the brighter with Harrad drilling a low shot wide in the second minute and Marlon Pack tested Barry Roche with a well struck free kick seven minutes later that was heading for the bottom left-hand corner.
The Shrimps hit back with Stewart Drummond showing a neat bit of skill to fire in a low volley from the edge of the box that forced Brown into an excellent save low to his right.
Cheltenham ended the half on top with Russell Penn seeing a shot from the edge of the box well parried by Roche before the Robins were denied a good penalty shout when Sam Deering went down under the attentions of Andy Fleming.