The police car, carrying two officers, was using its siren and lights, said Essex police.
The other vehicles involved in the crash on the A127 were a Fiat driven by a woman in her 70s and a BMW carrying a man, woman and three boys.
All the people involved were taken to Southend General Hospital.
The collision was being investigated by the Serious Collision Investigation Unit (SCIU), and the force's Professional Standards Department was made aware, said a police spokesman.
The Fiat driver had to be freed by the fire service after her vehicle left the road and ended up on a verge, police said.
But the five people in the black BMW, who are from Basildon, were not trapped, police added. The boys are aged five, seven and 11.
The two injured officers are both men, one in his 40s and the other in his 20s. They are based at Boreham.
The crash happened near the junction with Daws Heath Road just before 16:00 BST.
It forced the closure of the A127 near to where it had been shut because of flooding.