The accident, shortly after 7am, brought rush-hour traffic to a standstill in the Nab Wood area and blocked both sides of the road from New Close Road to Saltaire Road.
The crash involved a white Vauxhall van, a white BMW X1 and a blue BMW 520.
Firefighters from Shipley and Bingley were sent to the scene to rescue two people who were trapped.
They were transferred to hospital by ambulance.
Police said the casualties only sustained minor injuries.
Bus routes had to be diverted away from the scene.
The road reopened at 9.37am.