It appeared that the game was heading for penalties after Harry Winks and Vincent Janssen netted late goals for Mauricio Pochettino's team, but Marco Tumminello scored in the second minute of added-on time to earn the win for the Italian outfit.
A Tottenham Hotspur side which featured three of England Under-20s' World Cup heroes - Cameron Carter-Vickers, Kyle Walker-Peters and Joshua Onomah - should have fallen behind as early as the sixth minute, but Gregoire Defrel headed wide at the back post from around eight yards.
Spurs responded through Dele Alli, who fired straight at Alisson from inside the penalty area, but Roma went in front from the penalty spot as Diego Perotti placed the ball to Michel Vorm's left after Carter-Vickers was adjudged to have handled.
The Serie A giants soon began to seize control of the contest with Defrel nearly adding to their lead with a left-foot curler which went just wide of the far post, but Spurs still posed a threat at the other end of the pitch.
With eight minutes of the first half remaining, Harry Kane expertly controlled a long ball before his volley from 30 yards was well kept out by Alisson low down to his right.
Soon after the restart, Spurs should have been given a penalty when Spurs old boy Federico Fazio fouled Kane inside the area but the referee played on, much to the disgust of the England international.
It led to some heated exchanges - most of which involved Kane - but the officials just about kept a lid on things as Spurs began to show the qualities which helped them to second place in the Premier League table last season.
A neat passing exchange led to Alli having a shot deflected marginally off target, before Kane was denied a second goal of this American tour when his attempt from 12 yards was diverted wide of the far post.
Roma had been a non-entity in the final third but with 20 minutes left, they broke forward to net a second with Cengiz Under applying the finishing touches from eight yards after the ball had fallen invitingly for him from a miscued clearance.
However, with three minutes to go, Spurs started a dramatic conclusion to the match with a goal from Winks, who converted the rebound at the second attempt after Janssen had bunted the ball against the post.
It appeared too little too late for the North Londoners but in the first minute of added-on time, Janssen turned home at the near post after getting on the end of a cross from Georges-Kevin N'Koudou.
Penalties appeared to be a certainty but Roma went straight up the other end to regain the lead as Tumminello was well placed at the back post to slide in and convert from inside the six-yard box to seal a memorable triumph.