A number of children and adults were caught up in the "scary" incident at The Alphabet Zoo in Bradford on Tuesday afternoon.
Owner Ashiq Hussain said the man came "tearing through with the police right behind him" before jumping from a mezzanine level.
The police watchdog was informed of the incident as the man, who was the only person injured, received treatment in hospital.
West Yorkshire Police said it was called to the centre on Cemetery Road at 4.38pm.
The force said: "Officers attended and made continued efforts to detain the male but he then fell from a balcony.
"He was taken to hospital where his condition is described as serious but is not believed to be life-threatening.
"No-one else was injured during the incident
"Due to the nature of the incident it has been referred to the Independent Office for Police Conduct."
The centre provides play areas for children under 12 and babies and toddlers.
Mr Hussain said: "He's climbed over the secure door area, we don't let people through there without being allowed by staff, and he's run through the centre pushing children and adults out of the way.
"He's gone on to the first floor and, whilst he was going up there, he attacked a couple of customers, grabbing them by the throat and slapping them."
Mr Hussain said the police followed the man into the centre and tried to stop him but he broke free and jumped over the balcony crashing down into a fence which hit customers with children below.
He said: "It was scary. It's a safe place and people were enjoying themselves when this man comes tearing through with the police right behind him."
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