AFP reporters at the scene said chaotic scenes ensued in a packed square 10 minutes before the end of the match on Saturday evening, with the panic apparently triggered by fireworks being let off and one or more people shouting that a bomb had exploded.
Most of the injured were treated for cuts and bruises but seven had to be taken to emergency units at local hospitals.
As the scare spread through the crowd gathered in the city's Piazza San Carlo, a rush towards the exit points quickly accelerated and the square was evacuated so quickly it was left strewn with hundreds of sneakers ripped off people's feet as they ran.
The incident compounded a miserable night for supporters of Juventus, who lost the Cardiff final 4-1 to Real Madrid.
Some of the injuries occurred after a railing around the entrance to an underground parking beneath the square gave way under the weight of the crush.
Local media cited older Juventus fans present as saying the panic had evoked painful memories of the 1985 Heysel disaster, in which 39 mostly Italian fans died when fans were crushed by a collapsing wall before the start of that year's European Cup final, against Liverpool.