The penalty was awarded when Empoli's Piotr Zielinski slid in to make a clearance and seemed to get the ball away cleanly, but Roma forward Leandro Paredes tripped over his outstretched leg and referee Marco Di Bello pointed to the spot.
The official was surrounded by an angry group of Empoli players and both Matias Vecino and Zielinski were booked before De Rossi sent Davide Bassi the wrong way for the winner after the game had finished 1-1 at the end of 90 minutes.
Roma got off to a flying start when Juan Iturbe latched onto Maicon's cross and beat Bassi with an angled drive in the fifth minute but the Argentine did not celebrate the goal, apparently in protest at criticism of his recent performances.
Roma, second in Serie A, threatened to finish off the contest before halftime as they created a hatful of chances.
But Empoli stunned the Stadio Olimpico with 10 minutes left when Vecino's through ball sent Simone Verdi clear and he rounded Lukasz Skorupski to score into an empty net and force the match into the extra period.
Roma will face either Atlanta or Fiorentina, who meet in Florence on Wednesday, in the Coppa Italia quarter-finals.
(Writing by Brian Homewood; Editing by Ken Ferris)
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