Emergency services were called to Denby Dale Road near the junction with Thornes Road at around 1.30am on Saturday.
A black Mercedes Benz lorry hit the bridge and shed its load of paper and magazines onto the road.
While officers were dealing with the incident another lorry, travelling out of the city centre, also hit the bridge.
It took more than four-and-a-half hours to clear the road and examiners to assess damage to the bridge.
No-one was injured in the incidents.