The Iceland lorry ended up in the grounds of St John's Church following the crash on Woodfield Street, Morriston, at about 04:50 BST on Wednesday.
A spokeswoman for Iceland said the driver was taken ill at the wheel.
He was taken to Morriston Hospital "as a precaution", South Wales Police said.
Officers are investigating the cause of the crash, it added.
A delivery truck crashed into parked cars before hitting a church wall in Swansea.