The incident compounded a miserable night for supporters of Juventus, who lost the Cardiff final 4-1 to Real Madrid.
Thousands of people had gathered in Piazza San Carlo to watch Juventus face Real Madrid on giant TV screens.
Local reports have claimed that people were panicked when firecrackers were set off, prompting a surge in the crowd.
"At that moment, I saw the entire piazza went in the direction next to the screen to escape, all in a panic", said Brian Hendrie, an Associated Press reporter.
That was enough to trigger a stampede out of the city's Piazza San Carlo, in which the injuries occurred.
Many panic-stricken fans then started screaming and trying to run out of the place, triggering a large stampede. Two were reportedly in serious condition.
Afterwards shoes and bags littered the ground, people were seen limping and searching desperately for friends and relatives, the witness said.
For some Juventus fans, the incident brought back traumatic memories of the 1985 Heysel stadium disaster, where 39 people, majority Italians fans, were killed when a wall collapsed on them before the European Cup final against Liverpool.
Turin police say they are investigating the incident. Most of the injured were treated for cuts and light contusions.