Goalkeeper Wojciech Szczesny had an excellent debut for the visitors following his loan move from Arsenal, capped by a brilliant double save.
Edin Dzeko and Mohamed Salah, who are also on loan, drew a blank on their competitive debuts and it was left to Alessandro Florenzi to snatch a point for Roma who have finished second to Juve for the last two seasons.
“Dzeko was excellent in the first half and had that chance to win it with a header at the end,” said coach Rudi Garcia, starting his third season in charge.
“Dzeko and Salah need time to settle into the side but they’ve already done very well.”
Szczesny tipped away a Samuel Souprayen drive and was then rescued by Daniele De Rossi who headed Bosko Jankovic’s header off the line three minutes before halftime.
The Polish goalkeeper again excelled early in the second half to deny Emil Hallfredsson and Juanito Gomez.
Szczesny palmed away Hallfredsson’s long-range effort and then raced across his goal to get his hand to Gomez’s effort from the rebound.
Jankovic was not to be denied, however, as he volleyed Verona ahead from Hallfredsson’s cross in the 61st minute.
Florenzi replied five minutes later with a shot out of the blue which caught Verona goalkeeper Rafael off guard.
Dzeko headed over the bar in the closing minutes and Miralem Pjanic was thwarted by a superb stop from Rafael.
Elsewhere, Lazio beat Bologna 2-1, courtesy of a fluid start with two quickfire goals from Lucas Biglia and Ricardo Kishna, that put them into the ascendancy.
Matteo Mancosu pulled a goal back for the visitors before half-time but the hosts held on to record a win on the opening day of Serie A.