Humberside Police said officers had been called to reports of a collision at the junction of Ainsty Street and Hook Road shortly before 9am today.
"A woman passenger in the taxi suffered minor injuries in the collision and was taken to Scunthorpe hospital for treatment but has since been discharged," a spokesman said.
"The female driver of the Vauxhall car left the scene of the collision and was located and arrested later this morning. She has been taken into custody on suspicion of driving while under the influence of drink or drugs and for leaving the scene of a collision."
Anyone who witnessed the collision is asked to call the force on 101, quoting log 75 of November 28.