Blackburn loanee midfielder Morris, 22, proved he is a great catch when ran on to a slick pass from striker David Ball to smash in his first senior goal from the left side of the box in the 64th minute.
Delighted Fleetwood manager Graham Alexander saw his team move to within three points of an automatic promotion place and seal a play-off place, but admitted:
"We put ourselves on the back foot in the first half and had to dig deep to get the three points. We didn't rant and rave at half time, just talked to the players calmly and I thought the response was great. They deserved the win."
Veteran striker Jamie Cureton, 38, brought the game to life in the 32nd minute with his ninth goal of the season.
Midfielder Harry Williams saw keeper Chris Maxwell do well to get down and save his low strike from eight yards, but Cureton pounced to fire in the rebound from the left side of the six yard box.
Midfielder Antoni Sarcevic pulled Fleetwood back into the game from the spot in the 55th minute.
Robins midfielder Matt Richards hauled down Iain Hume as the Preston loan striker darted into the right side of the box and Sarcevic blasted in his fourteenth of the campaign.
Maxwell played a big part in the win when he pulled off two great second half finger tip saves.
Defenders Sido Jombati and Steve Elliott both had Maxwell dive at full stretch to keep his team ahead as Cheltenham pressed for an equaliser.
Furious Cheltenham boss Mark Yates groaned:
"We seem to find ways to lose games. Words fail me. We've done what we've done well all season and thrown away the lead."