On a tense evening, both sides were high on effort but often low on quality, although there was enough goalmouth action at both ends to keep the crowd entertained.
Cheltenham secured their third 1-0 win in the last four matches to keep their hopes of avoiding the drop alive, although things now look increasingly tough for bottom of the table Port Vale.
Cheltenham made a good start with winger Michael D'Agostino carving a path down the right wing.
The on-loan Blackpool youngster crossed for striker Connor to head onto the top of the net with only eight minutes played.
But 14 minutes later he was involved in the only goal. Steven Gillespie carried the ball out of defence and found D'Agostino wide on the right.
Gillespie then collected a return pass at speed and crossed low for Connor to touch in at the near post.
Cheltenham almost doubled their lead a minute later when Vale goalkeeper Joe Anyon got down to save well from Gillespie, but the balance of the game shifted as the first half wore on.
Cheltenham's passing became ragged and their goalkeeper Shane Higgs was forced to make a flying save from Justin Miller - a reflex block from Danny Whitaker and a superb flying stop from a Marc Richards header.
Vale's woes were compounded midway through the second half when Richards was carried off on a stretcher.
His replacement, Joe Cardle, then saw a shot parried by Higgs but only as far as Danny Glover, whose close-range effort was blocked by a brave last-ditch tackle from Alan Wright.
Gillespie should have made the points safe in injury-time when his low drive was saved by Anyon, while Higgs was called upon to make one more reflex save, this time from Glover in the dying seconds of the contest.