Crewe, all clever flicks and crisp passes, looked good but failed to find the cutting edge needed to end a run of five games without a win.
Cheltenham, meanwhile, may be in the relegation zone but if they were given points for effort and endeavour they would be challenging with the leaders.
Crewe forged ahead in the 15th minute when a clever build-up ended with a beautifully accurate through pass from David Vaughan to Luke Varney.
The highly-rated frontman took a touch before driving a low shot beyond goalkeeper Shane Higgs.
The Cheltenham goalkeeper needed to be at his sharpest to save another well-struck shot, this time from Vaughan, as the Alex turned on the style with some crisp passing during a bright first-half spell.
In fact, Cheltenham had offered very little as an attacking force until they came up with an equaliser on the half-hour mark.
Brian Wilson's free-kick from the right-hand corner of the area bounced back off the defensive wall and fell to captain John Finnigan 18 yards out.
Finnigan was afforded enough room by the Crewe defence to strike a shot beyond the outstretched arms of goalkeeper Ben Williams and just inside his right-hand post.
The second half proved an absorbing 45 minutes that could have gone the way of either team.
Cheltenham had more of the ball but Crewe had the better chances with Varney sending a firm volley just over the bar and defender Billy Jones heading inches wide from a Ben Rix corner.
The best chance of the second half fell to the visitors when substitute Lee Matthews broke into the box but Higgs got down to parry his shot with a smart save.
Cheltenham's Melligan almost won it with a low shot that brushed the post in the closing minutes but a defeat would have been harsh on either set of players.