Police in Cleveland say the car chase started in the suburb of Garfield Heights - after the man committed a traffic violation - and reached speeds of 145km/h.
Officers say that after a race through several communities, the driver and a passenger bolted from the car and jumped over a fence.
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But they apparently did not realise the barrier was on the outside of the state women's prison.
The pair were arrested along with two other passengers in the car who also tried to flee.
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