The man was at the wheel of a silver-coloured Vauxhall Astra that failed to stop after it was spotted travelling at speed in Harehills Road, Harehills, by officers in a marked police vehicle at about 12.20am today.
A spokesman for West Yorkshire Police said the Astra was pursued around the Harehills area for the next few minutes before being abandoned in Strathmore Terrace.
The driver then ran off and, despite a subsequent search by officers aided by the National Police Air Service helicopter, managed to evade capture.
He is described as Asian, with a beard, and was wearing a blue hooded top and shorts.
Police say the man headed off in the direction of Morrisons after dumping his car.