The Giallorossi went behind twice at the Stadio Olimpico - once in each half - but showed resilience to bounce back quickly both times, as first Zaniolo headed home to restore parity following German Pezzella's early finish.
Then, in the second half, Perotti made it three goals in as many games as he lashed home from Justin Kluivert's cross - erasing Gerson's deflecting strike and setting up and exciting final period in the Italian capital.
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The Giallorossi pushed for a winner but it proved elusive, with Edin Dzeko and Steven Nzonzi among those to go closest in the final 30 minutes.
The result means Roma remain winless in their last three games, and know they need to turn things around on Saturday against Sampdoria if they are to aid their chances of finishing in the top four and qualifying for next season's Champions League.
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