The item had been dropped off in an envelope in late July.
Earlier this month, East Bristol Auctions estimated the glasses would be sold for around £10,000 to £15,000.
However, they have now gone for £260,000.
The auction house posted a video on social media of the moment the glasses were sold for that price.
“We found them just four weeks ago in our letterbox, left there by a gentleman whose uncle had been given them by Gandhi himself,” they wrote on Friday.
“An incredible result for an incredible item! Thanks to all those who bid.”
Andrew Stowe, an auctioneer, explained to Sky News earlier this month how his Bristol auction house had come across a pair of the assassinated independence leader’s glasses.
He said a man had dropped them off in their letterbox on Friday 31 July, but staff only picked them up when they came back into work on the Monday after.
“A colleague of mine picked them up, ripped open the envelope and found a brief note inside saying, ‘These glasses belonged to Gandhi, give me a call’,” he explained.
Mr Stowe said the glasses were “just hanging out of our letterbox” when they were discovered.
On the listing on the auction house website, it said the glasses were given to the vendor’s uncle by Gandhi in South Africa on a visit.
“The uncle [was] working for British Petroleum at the time and was stationed in South Africa, and it can be presumed that these were gifted by way of thanks from Gandhi for some good deed,” it said.