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Roma dominated the opening half-hour, while the visitors had to wait until the 34th minute for their first shot, when Ciro Immobile sliced the ball wide.
Padelli was tested again before the break as Miralem Pjanic looked for the opening goal, but Destro was able to break the deadlock just a couple of minutes before half time when he slotted in from a Gervinho delivery into the box.
The visitors were back on level terms early in the second half when Immobile tapped in from close range following a cross from Giuseppe Vives.
Roma looked to restore their lead quickly, as Pjanic forced a save before Radja Nainggolan headed wide just minutes after Immobile's goal.
Both sides has chances for late winners, as Gervinho blasted the ball wide for Roma, while Alessio Cerci would not keep his effort on target.
However, the hosts kept pressing into stoppage time, and were able to restore their lead with just moments to spare when Gervinho drilled the ball in for Florenzi to blast into the top corner, sealing the win for the hosts.