Jean-Louis Akpa Akpro’s first-half goal was enough to take the Shrews up at Cheltenham, whose relegation to the Conference was confirmed after 16 years in the Football League.
“It’s nice to finally say we are promoted,” Mellon said. “It’s a terrific effort since day one of pre-season.
“We’ve always had the mindset of trying to bounce this club and I am delighted we’ve done it here.
“It’ll take some time to settle in. It was a tough game and we knew they’d put us under pressure, but we stood firm to a man and battled through and to get an away win is always hard so it’s a terrific effort from the boys.”
Mellon was delighted to see Akpa Akpro net the winner: “It’s a nice moment for him, finally!
“We are all fond of him so for him to get the goal I am delighted for him, for the chairman and for many others who have inspired us to do this.
“Our fans were magnificent and we’ve said for the lads to enjoy it because they deserve it.
“They’ve roared us on and it’s a memorable day, one which will stay in our heads for a long time.”
Cheltenham boss Gary Johnson said he felt let down by some of his players, who he said lacked the professionalism required to get out of trouble.
“The dressing room is in bits,” Johnson said. “It’s always very sad when a team are relegated, especially out of the league.
“Unfortunately when you are relegated you generally deserve to be down there, for whatever reason. Unfortunately there hasn’t been enough in the team to keep the club in the league.
“We’ve tried very hard, working all hours God sends and we’ve tried to get the players doing the same, but the losers among them are less than professional.
“I’ve had to tell a few home truths to people, some we’ve trusted and some we haven’t, but we’ve had to use them and they end up letting you down.
“Something is not right, recruitment, and there is a fitness level and some of them found it very hard to do that. That’s what I found was more disappointing than anything else. The injuries come in when you need people and they’ve not been available for certain reasons.
“I’ve given them the benefit of my experience and told them some of them will end up looking for new clubs each year.”
Mellon’s men took the lead in the 20th minute when striker Akpa Akpro’s low shot took a heavy deflection off Durrell Berry, giving Trevor Carson no chance.
Cheltenham missed a good chance seven minutes later when Wes Burns headed Matt Richards’ cross wide after a short corner taken by Matt Sparrow.
Shaun Harrad went down in the box under a challenge from Jermaine Grandison in the 38th minute, but a free-kick was given against him and Shrews goalkeeper Jayson Leutwiler denied Burns in first half stoppage time.
Cheltenham’s first real attempt of the second half came from goalkeeper Carson, whose huge clearance had to be turned over the bar by his opposition number.
Carson saved a low effort from James Collins in the 65th minute and while Harrad went close with a free-kick and Zack Kotwica saw a drive well blocked by Mark Ellis, Cheltenham did not have enough to level.