The veteran striker was shown a red card in the 34th minute for a kick on Emerson Thome to leave his side facing a rearguard action.
Gareth Ainsworth had given his side the lead five minutes earlier but once Furlong was sent off the Londoners were forced to bring down the shutters.
They defended in depth and a breakaway goal from Kevin Gallen appeared to have sealed victory until Dexter Blackstock scored a debut goal for the Rams four minutes later.
But Royce, capped his penalty save from Inigo Idiakez with an even better stop to leave Derby uncomfortably close to the drop zone.
Rangers showed plenty of character after Furlong's moment of madness and Ian Evatt marked his return to his former club with a towering performance.
Derby hammered away throughout the second half but Rangers soaked up the pressure until Lloyd Dyer handled in the 73rd minute.
Idiakez checked his run up and Royce guessed the right way and beat the ball to safety as he dived to his right.
When Gallen curled a brilliant shot inside the far post after Georges Santos had put him in on the Derby right, the game looked over but Blackstock pounced from close range four minutes later.
When the on-loan Southampton striker set up Idiakez a minute later an equaliser looked a certainty, but Royce managed to save one handed as he fell backwards.
It was a remarkable piece of goalkeeping and left Derby with only one win in their last 12 games.