Yeovil went in to the match on the back of four successive victories, while Cheltenham's six straight defeats suggested there could be only one result.
The visitors had other ideas and Robins boss John Ward saw his team snatch a victory which on the balance of play they deserved.
Yeovil manager Russell Slade must be fed up with Ward. They met at the end of last season in the League Two play-off final, when Slade was boss at Grimsby Town, and Cheltenham also emerged victorious on that occasion.
What made Cheltenham's victory even more commendable was that they were without five key players, including skipper John Finnigan who was completing a three-match ban and Grant McCann who was on international duty with Northern Ireland.
But visitors took the fight to Yeovil and the Glovers' Steve Mildenhall was by far the busier of the two goalkeepers in the first half.
He had to be alert to tip over a shot from Gavin Caines that took a deflection on 20 minutes and, nine minutes later, was helpless when a drive from Brian Wilson also hit a defender and flashed wide of the right-hand post with the goalkeeper stranded.
Yeovil came back in the second half and were unlucky on 47 minutes when Wayne Gray hit the top of the bar.
Two minutes later the same player went past three men on the touchline to set up a chance for Marcus Stewart, but the experienced striker slipped at the critical moment.
Cheltenham enjoyed a spell of sustained pressure late on and Steve Guinan, who had been anonymous up until that point, hit a post and saw two other shots go wide.
But with two minutes of normal time remaining the Robins got the goal.
Mickey Bell provided a perfect cross to the far post from way out on the left and Melligan stooped to head in.
In the three minutes of stoppage-time, Yeovil's Paul Terry almost equalised with a terrific low drive that brought the save of the night from Scott Brown, but Cheltenham held on to give themselves a welcome lift.