By Agence France-Presse
Most of those hurt, one seriously, suffered leg injuries after getting entangled in the escalator's mangled mechanics, Italian media added.
A video of the drama shows the descending escalator at Repubblica station suddenly and dramatically pick up speed with dozens of people on it, many screaming as they piled on top of one another at the foot of the moving stairway.
Media reports said witnesses spoke of football fans, seemingly drunk, jumping and dancing on the escalator just before its collapse, but supporters denied this ever happened.
"In any event, we are here to understand what happened and to offer our support to the wounded and their families," Rome mayor Virginia Raggi told journalists at the scene.
Several investigations have been opened, and the metro station near Rome's Termini train station was closed.
The horrific events took place around 1730 GMT, just over an hour before the kickoff of a Champion's League match between Italy's Roma and Russian club CSKA Moscow.
The Russian fans were headed for the metro that would have linked them to the train to the stadium.
There was heightened security in Rome for some 1,500 Russian supporters, notoriously rowdy, expected to watch the game.
One Russian supporter, the subject of a stadium ban, had already been stopped at the airport as he tried to enter.
But according to Italian media, this did not prevent an assortment of incidents around the stadium hosting the match. One Russian fan was stabbed with a knife, and two others were hurt in clashes with other supporters.
Roma won the group stage match 3-0, tweeting after the match that: "The thoughts of everyone at #ASRoma are with the supporters injured prior to tonight's game. The club will do everything it can to provide the right support and assistance to those in need."
Rome's public transport is often criticised as running on obsolete equipment.
In recent years, more than a dozen municipal buses have caught fire while in service, with several reports of metro passengers injured by malfunctioning hardware.