The comedian presented Sainsbury's worker Eddy, from Glasgow, with the money on stage at one of his comedy shows after he took part in Carr's quiz show I Literally Just Told You.
During the show Eddy was asked to recall how old the pop star Ariana Grande is.
Contestants are meant to be told the answers to questions throughout the show and need to rely on their memory to come up with the correct answer at the right time.
However, Grande's age, which is 28, was never actually mentioned during I Literally Just Told You.
In order to rectify the mistake, Channel 4 invited Eddy to one of Carr's gigs at the O2, where he was invited on to the stage by the comedian.
Carr told him: "I've got something to tell you. I've got something to confess. You lost the money on a question - it was Ariana Grande's age.
"We talked about her age, but we never gave you the specific age.
"So we messed up and when you mess up in life, you've got to own your mistakes. That's a cheque for £18,000."
Eddy was in disbelief and screamed "Oh my God" as he was handed the cheque by Carr.
"Eddy, I've got a question for you... You've just won £18,000, but how do you know that?" continued Carr.
"Because Jimmy, you literally just told me so," replied Eddy.
The next episode of I Literally Just Told You airs at 10pm on Wednesday January 12 on Channel 4.