The Shrews struck nine minutes after the break to register a famous victory as they knocked out a side two divisions above them at New Meadow.
"I'm so proud of my players," Mellon told BBC Sport after the win. "This was no smash and grab. We worked hard, played well and thoroughly deserved to win.
"We don't just lump it in the box. We get the ball down and knock it about and I don't remember our keeper having to make a save.
"I've said since I first came in that my first job was putting a smile on the fans' faces after a year of doom and gloom."