Kris Commons was the star of the show with an impressive hat-trick, but his team were all-round way to strong for the struggling Robins.
Forest started strongly and it took them only 11 minutes to take the lead.
Chris Cohen played the ball out to left-back Julian Bennett and his in-swinging cross was headed in by Commons, who had made a clever run across the box.
Cheltenham's Dean Sinclair had the ball in the net shortly afterwards but the goal was ruled out for offside and Forest looked sharper than their hosts all across the pitch.
Northern Ireland international Sammy Clingan had a shot blocked as the visitors dominated the opening stages.
Cheltenham's former Nottingham Forest left-back Alan Wright saw a free-kick from the edge of the box saved at full-stretch by Forest goalkeeper Paul Smith, but the game was all but decided in the moments before half-time.
Grant Holt sent a header inches wide from a cross by Clingan, but only seconds after that, Commons darted through a big gap in the heart of the Cheltenham defence before drilling a low shot beyond goalkeeper Shane Higgs.
Clingan saw a shot cleared off the line by defender Andy Gallinagh at the start of the second half, but Commons completed his hat-trick after 66 minutes.
A sliced clearance from Higgs struck Forest's Grant Holt and spun across for Commons, who hit the ball first time from 35 yards into an empty net to make it seven games unbeaten and move his team up to third place in the table.
Cheltenham, meanwhile, will be hoping caretaker boss Keith Downing can inspire his charges to pick up points before a decision is made on whether he will get the job on a full-time basis.