Trailing 2-1 going into the final 10 minutes, Gian Piero Gasperini decided to bring on the Australian-born centre forward who has represented Italy in underage football since last year.
Earlier, a Che Adams brace either side of a strike by Mario Hermoso had given Torino the advantage.
On his debut for Roma, Arena sent the Stadio Olimpico wild as his first touch of the ball was to head in a leveller for the hosts.
However, Roman joy was short-lived as Emirhan Ilkhan netted a 90th-minute winner for Torino.
Torino will face Serie A leaders Inter Milan in the next round.
Roma's elimination compounds their recent struggles. The Giallorossi, who briefly took the lead in Serie A after the 12th matchday, have lost four of their last eight league games as they currently sit in fifth place.
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