Tumminello's winner came just a minute after Spurs' Vincent Janssen had tied the match by scoring from a scramble in front of goal. Harry Winks had brought the Premier League club back into the match with his 87th minute goal after Roma had taken a 2-0 lead through Diego Perotti and Cengiz Ünder.
Referee Hilario Grajeda handed out only two yellow cards in a match that featured several heavy tackles and some pushing and shoving from two sides who finished second in their leagues last season. Spurs manager Mauricio Pochettino also came close to being sent to the stands.
Roma coach Eusebio Di Francesco told reporters his side would be buoyed by the win but did not read too much into the result.
"We played a good performance but it's still July. It's too early to make assessments," he said via an interpreter.
"Obviously it's a boost of confidence for me and the lads."
Roma opened the scoring in the 13th minute through Perotti's penalty kick after Spurs defender Cameron Carter-Vickers was ruled to have handled in the box.
Turkish 20-year-old midfielder Ünder, who signed a five-year deal from Basaksehir this month, doubled Roma's advantage with 20 minutes left.
Spurs coach Pochettino said his side were still trying to find their way back to form.
"I think we are working a lot to try to develop our game and improve. We have to fight for everything," he added. "But I am calm and I am happy.
"To win is not the most important thing. It is to get fit and try to improve in every way."
(Editing by Peter Rutherford)
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