Christian Lealiifano was awarded a controversial try late in the first half from what appeared to be a forward pass, while replacement prop Allan Alaalatoa had his second half try cancelled out after a knock-on was discovered in the build-up.
Lealiifano's try was allowed to stand because, facing a barrage of boos from the crowd who saw a replay of the final pass from James Dargaville, the flyhalf took a drop kick for the conversion to ensure that referee Rohan Hoffmann could not ask for the try to be reviewed.
But he did not act as quickly when Alaalatoa took a pass from fellow front rower Ruan Smith and produced a lovely left-foot step around the final defender with about 15 minutes remaining.
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Hoffmann this time asked for the build-up to be reviewed and decided that blindside flanker Scott Fardy had lost the ball forward in a tackle, ruling out the prop's spectacular try.
Fardy's fellow loose forward David Pocock, lock Rory Arnold and winger Nigel Ah Wong also scored tries for the visitors.
Lealiifano's normally reliable goal kicking was off target as he slotted just a penalty and conversion for the Brumbies, who moved to 29 points on the table, four ahead of the New South Wales Waratahs who play on Saturday.
Rebels winger Cam Crawford had scored his side's opening try, while Jack Debreczeni added a conversion and penalty as the home side went into the break trailing 20-10.
Rebels inside centre Reece Hodge also scored a try following a sustained period of attack. Debreczeni's conversion narrowed the gap to three points.
Melbourne winger Sefa Naivalu was the beneficiary of a poor read by the Brumbies defence and he went over untouched following an attacking lineout.
Ah Wong completed the scoring with about 10 minutes remaining.
(Reporting by Greg Stutchbury in Wellington; Editing by Mark Trevelyan)
