Ervin Zukanovic, another new signing, also played a key role in Roma's first win in three attempts under Spalletti, who was brought back for a second stint at the club after Rudi Garcia was sacked earlier this month.
Zukanovic, signed on loan from Sampdoria, prevented Roma from going behind when he headed the ball off the line in the 17th minute.
It was a crucial save as one minute later Radja Nainggolan, despite being on the ground in the middle of a penalty area scramble, managed to poke the ball past Nicola Leali to give Roma the lead.
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Daniel Ciofani brought Serie A debutants Frosinone back into the game when he cleverly flicked the ball up and volleyed past Wojciech Szczesny five minutes later.
El Shaarawy, signed on loan from AC Milan, put Roma back in front early in the second half when, with his back to goal, he managed to flick Zukanovic's low cross into the goal with his heel from 12 metres.
Stalwart Francesco Totti came on in the 59th minute for only his second league appearance since recovering from a thigh injury and the 39-year-old set up Miralem Pjanic for the third six minutes from the end.
Roma stayed fifth with 38 points, nine behind leaders Napoli who are at home to Empoli on Sunday.
(Writing by Brian Homewood in Berne, editing by Alan Baldwin)
