Conte seemed to have already given Tottenham a much-needed spark as his team led 3-0 after just 28 minutes, but Vitesse pulled two goals back before halftime.
Roma coach Jose Mourinho was fined 10,000 euros ($11,500) for “disrespectful” behaviour toward the referee after the game.
AS Roma is currently at the fourth position in Serie A standings with 15 points from seven games.
Roma are Italy's representatives and Jose Mourinho's side came from behind to thump CSKA Sofia 5-1.
Jose Mourinho said he felt like a child when charging from his dugout to celebrate Stephan El Shaarawy's last-gasp winner in a thrilling 2-1 victory over Sassuolo.
Nicolo Zaniolo and substitute Stephan El Shaarawy both scored in the second half as Roma progressed with the minimum of fuss.
Allegri returns to the scene of his greatest triumphs after two years out of the game, and is one of three big coaching names to retake their place on the sidelines.
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.
Shaw suffered at the hands of Mourinho for three seasons at Manchester United, a string of public criticism not a patch on what was said behind closed doors, the 25-year-old admitted this week.
The England international was on the pitch for just 233 minutes under Mourinho in the 32 league matches, costing him a place in the England squad for Euro 2020.
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.
AS Roma coach Jose Mourinho has said England should be able to reach the finals of the ongoing Euro 2020.
Igor Stimac has said even Pep Guardiola or Jose Mourinho could not do better than him in the current circumstances with the Blue Tigers.
The only club that the 58-year-old failed to leave his impression upon and departed empty-handed was the North London's Spurs.
Earlier, Spurs had made Leicester manager Brendan Rodgers their top target after Leipzig's Julian Nagelsmann agreed to take the Bayern Munich post.