Celtic supporters celebrated their penalty shoot-out victory in the quarter-final at Ibrox by pouring onto the pitch, sparking a retaliatory invasion from hundreds of Rangers fans.
Police and stewards eventually formed a barrier between the rival fans after missiles, including a lit flare, were thrown.
Police said a 10-year-old child was hit by a coin and one officer was struck on the head by a bottle.
Footage showed a female officer being kicked to the ground from behind by a Rangers fan who was sprinting towards the Broomloan stand, which held 7,500 Celtic supporters.
A 47-year-old man was apprehended by police after an incident involving a member of the Celtic coaching staff.
Mobile phone footage also showed a fan aiming a kick at Celtic defender Julian Araujo, before another ran up and pushed the Mexico international.
Police had to deal with incidents outside the stadium, with several officers, stewards and members of the public injured in the chaotic scenes.
Nine arrests have been made inside 24 hours, with "significantly" more expected, prompting calls to reduce the away allocations when the Glasgow rivals meet.
Celtic were allowed 7,500 fans inside Ibrox for the first time since 2018.
Recent away allocations have veered from about 750 to a complete absence to around 2,500, but Scottish Cup rules state away clubs can claim up to 20 per cent of tickets and a deal was reached following input from the Scottish Football Association.
However, Police Scotland Chief Superintendent Emma Croft believes the allocation had an impact on the trouble.
"There's the ongoing rivalry and pent-up aggression between the two sides, absolutely," she said.
"And for me, also, such a large away fan base within Ibrox, which was news to many people and a new experience for many people, given the time, I think played its part, absolutely.
"I don't think it's right that I say that it's a complete ban, but what I would say is that the set-up is not sustainable and not safe."
A number of Celtic fans without tickets broke into the stadium and graffiti in the Broomloan Stand included references to the 1971 Ibrox disaster, when 66 Rangers fans died in a crush as they left an Old Firm game.
A review of the police operation has been launched but Croft feels enough officers were deployed and that they reacted "as quickly and appropriately as we possibly could".
"We are dealing with threat and risk in many areas of the stadium, inside and outwith. The public see one area and that's where the cameras pan to," she said.
"It's very difficult to respond on the field of play when you have a number of other people who are intent on trying to meet together to cause violence."
