It is thought Harnham Gate was hit by a van at around 2am on Saturday.
Cathedral officials are concerned the repairs could prove costly and take months to complete.
The wooden gates are believed to be around 80 years old and the stone work surrounding them dates back to the 14th century.
Expressing his disappointment, the Cathedral's clerk of works, Gary Price said: "The gates and the stone work are very old and it will take a lot of work to fix them.
"When I first heard about the gate I thought that someone had maybe just knocked into them.
"But when I got there and saw for myself I was shocked - I wasn't expecting to the see the carnage that I did and the gates hanging off their hinges.
"It looked as if a herd of elephants had come crashing through them."
Wiltshire Police said that a man, 51, has been arrested on suspicion of failing to stop, criminal damage and driving with excess alcohol, and has been released on bail pending further inquiries.
Police are asking anyone with information to call 101 or to contact Crimestoppers anonymously on 0800 555 111.
