But the former Wolves midfielder's first-half header was clouded in controversy with the Shakers insistent he was offside when he headed home Brian Wilson's cross at the back post three minutes before the interval.
Simon Whaley, who turned down a loan move to Sheffield Wednesday on Thursday's transfer deadline day, nearly gave Bury the lead in the first minute. The midfielder was teed up by loan Wednesday striker Danny Reet, but his powerful shot thundered into the crossbar.
Reet, who scored twice on his Shakers debut, then had a good chance of his own. He chased Colin Woodthorpe's punt forward, but fired into the side-netting from a tight angle.
Bury pressed further and Reet set up for Whaley for a second time, this time the midfielder saw his side-footed shot saved by Cheltenham goalkeeper Shane Higgs.
Cheltenham's first foray on goal came when striker Kayode Odejayi, back after injury, clicked the ball across the six-yard box, but midfielder Brian Wilson saw his shot saved by Bury keeper Neil Edwards.
Steve Guinan beat the offside trap as the game swayed into Cheltenham's favour, but his cut back was behind the on-rushing Wilson, who would have surely scored from close range.
Melligan's corner found Shane Duff midway through the half, but his close-range header was brilliantly saved by Edwards and then Tom Kennedy was forced to clear off his own goal-line when Duff directed the rebound goalwards.
Kennedy delivered a testing free-kick for Cheltenham's defence a minute later but, although Dave Challinor headed into the path of Tom Youngs, the centre-forward scuffed his shot.
Grant McGann saw a long-range shot well-saved by Edwards and then, as the half drew to a close, fired wide of the far post with a left-foot strike.
But the deadlock was broken three minutes before the interval when winger Wilson crossed to the back post, where a willing and waiting unmarked Melligan was only too happy to head home.
Second-half substitute Brian Barry-Murphy nearly scored an own-goal having only been on the pitch for two minutes, but having miscued horribly from Melligan's corner, Edwards was luckily to the rescue.
Fellow substitute Jake Speight had a long-range shot saved by Higgs as Bury pressed for an equaliser and Whaley also saw an effort saved from distance.
At the other end Craig Armstrong nearly doubled the visitors' advantage, but his curling shot flew inches wide of the far post.