Nicolo Zaniolo and substitute Stephan El Shaarawy both scored in the second half as Roma progressed with the minimum of fuss.
Abraham Tammy, who has played six times for England, helped Chelsea win the Champions League last season.
At his first news conference as Roma coach on Thursday, Mourinho's steely determination and prickly nature resurfaced when questioned about his apparent slide away from the coaching elite.
Despite the heat they thronged outside the city's Ciampino airport to greet the self-declared 'Special One' with a party atmosphere and flares heralding in the new era.
Chris Smalling and his family were held up at gunpoint inside their home in the Italian capital in the early hours of Friday morning.
In a country known for boisterous soccer club presidents, the new American owners of Roma are bucking the trend with the silence treatment.
Inter Milan -- fourth last season -- suffered an early upset after Domenico Berardi slotted in a 27th-minute penalty to give all three points to their rivals from Modena.
Football Transfers: French Ligue 1 side Bordeaux block Malcom move to AS Roma after late Barcelona bid
The 21-year-old had been expected to arrive in Rome on Monday night for a medical on Tuesday before joining his future teammates on their pre-season tour of the United States.
The 29-year-old midfielder Javier Pastore signed a five-year contract with AS Roma that runs until June 30, 2023.
Spanish veteran defender Ivan Marcano has signed for Serie A giants Roma, the Italian side confirmed on Thursday.
Coric made his senior debut for Dinamo in 2014, and went on to win four Croatian League titles and three Croatian Cup trophies there.
Totti now has a director's role with the Italian club, who this season reached the Champions League semi-finals for the first time in 24 years.
The Paris-born Algeria forward is likely to command a fee of around 50 million pounds, with Roma's offer believed to be some way short of that figure.
Edin Dzeko scored twice to help Roma to a 4-1 victory at AC Milan on Sunday and keep the capital side's title chances mathematically alive.