David Wotherspoon's deflected effort in the 82nd minute secured victory for Hibs, who had not beaten the Jambos in their previous 12 meetings, including a 5-1 drubbing in the final of last season's competition.
"In cup football you take what's going, we showed great character today," said Fenlon.
"There wasn't loads of football played in the game but I'm delighted for the supporters. They're the people that deserve it today.
"They can go away with their heads held high. They've been down in the doldrums for too long, to be honest, and that was for them."
Despite being delighted with the result, Fenlon did not feel it was a reaction to losing last season's cup final to Hearts.
"I don't think so, that's got nothing to do with it," he added.
"It's about rebuilding this club and getting us back to where we should be."