Manchester police blamed excessive alcohol consumption for "isolated pockets of disorder" among thousands of Rangers fans at public screenings of the game. The stabbing occurred outside the City of Manchester Stadium before the game. Police said he had non-life threatening injuries.
Police said the day was "largely peaceful" despite several police officers and fans being injured during skirmishes.
Zenit won the match 2-0, but Russian fans were far outnumbered by Rangers supporters in designated fan zones in downtown Manchester, with an estimated 100,000 ticketless supporters mingling in the city before watching the match live on giant screens.
Violence broke out when one screen broke down 15 minutes before kickoff.
"As a result, a small number of supporters - who until that point had been largely well-behaved and had been enjoying the carnival atmosphere - became frustrated and started to pelt the screen with beer bottles," Manchester Police said in a statement.
"Council workers attempted to repair the screen, however these attempts failed and supporters, many of which had been drinking heavily for a number of hours, became increasingly frustrated."
Buses were arranged to transport fans to a screening at a velodrome next to the stadium, but "a significant minority" remained at the site and began to confront police.
"A number of people were arrested during the disorder," the statement said. "A number of people, including police officers, were injured as officers brought the disorder under control."
The police said the "vast majority of supporters have behaved impeccably." It said there was no other trouble in other screenings in the downtown area.
Until the screen broke down, there had been eight arrests.
"I am sure the majority of Rangers fans will be as disappointed as we are that this event was marred by this unnecessary and unacceptable behavior by a small number of fans," assistant chief constable Justine Curran said.
"Our priority has been public safety and my officers have acted professionally, swiftly and decisively to bring this trouble under control."
Of the other arrests, four were for public disorder offenses, two for ticket touting, four for affray - or starting a brawl, one for possession of a weapon, four on various assault charges, two for possession of drugs, three for theft, two for forged tickets and two for invading the field.
Outside Rangers' Ibrox stadium in Glasgow, Scotland, which hosted 25,000 fans for a public screening of the final, Scottish police said 21 people had been arrested on public disorder offenses.
