The previous record was Gerry Hitchens' 16 goals for Inter Milan in 1961-62.
Roma finish in sixth place and qualify for next season's Europa League.
"It's been a good year for me," Abraham said. "I hope to build on this year and hopefully we can finish with a trophy and it can be the perfect year."
Roma, who play in the Europa Conference League final on Wednesday against Feyenoord, took the lead at Torino after 33 minutes as Abraham turned and finished inside the penalty box after a pass from captain Lorenzo Pellegrini.
Abraham doubled their lead just before half-time with a penalty, before a late spot-kick scored by Pellegrini sealed the success.
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