I don't recall the Presidents of Harvard & Yale resigning over "W's" degrees
Sir Howard Davies said he recognised the university's reputation had "suffered" and he had to quit.
He said the decision to accept £300,000 for research from a foundation run by Col Gaddafi's son, Saif, "backfired".
The LSE council has commissioned an independent inquiry into the university's relationship with Libya and Saif Gaddafi.
It will seek to clarify the extent of the LSE's links with Libya and establish guidelines for future donations.
Lord Woolf, former Lord Chief Justice of England and Wales and former chairman of the Council of University College London, has been appointed to carry it out.
Sir Howard said he regretted visiting Libya to advise its regime about financial reforms, calling it a "personal error of judgement".
"I have concluded that it would be right for me to step down even though I know that this will cause difficulty for the institution I have come to love," he said.
"The short point is that I am responsible for the school's reputation, and that has suffered."
He also said he had advised that it was "reasonable" to accept the money, which turned out to be a "mistake".
There were risks involved which should have been weighed more heavily in the balance, he concluded in his resignation letter.
Sir Howard is a former head of the Financial Services Authority and deputy governor of the Bank of England.
He will remain as the head of the LSE until a successor has been found.
Peter Sutherland, chairman of the LSE's court of governors, said Sir Howard had been an "outstanding" director over the past eight years.
"We accept his resignation with great regret and reluctance but understand that he has taken an honourable course in the best interests of the school," he said.
The LSE has already announced it is investigating claims that Saif Gaddafi plagiarised his PhD thesis, which was awarded in 2008.
The Libyan leader's son had studied at the LSE, gaining both an MSc and PhD.