By Sunday morning, 3,048,000 people had signed the petition on the official UK Parliament website.
That number takes it well over the 100,000-signature threshold needed to force a debate on the issue by members of Parliament.
A rush to sign the petition caused the website to crash temporarily due to the high volume of traffic Saturday.
The petition, set up by William Oliver Healey, states: "We the undersigned call upon [the UK] Government to implement a rule that if the Remain or Leave vote is less than 60%, based on a turnout less than 75%, there should be another referendum."
Thursday's referendum had a turnout of 72% -- an increase over last year's general election turnout of 66%, but below the 75% suggested in the petition.
The "Leave" campaign won with 17,410,74 votes -- 52% -- to the "Remain" team's 16,141,241, or 48%.
Ben Howlett, a Conservative MP, confirmed on Twitter t
han the petition would be discussed by the House of Commons petitions Select Committee
That news came as some voters who had backed the "Leave" campaign took to Twitter to register their regret -- adopting the #Regrexit hashtag.
One voter, Adam from Manchester, told the BBC: "My vote -- I didn't think was going to matter too much because I thought we were just going to remain.
"The David Cameron resignation has blown me away to be honest. I think the period of uncertainty we're going to have for the next few months has been magnified, so I'm quite worried."
Another "Leave" voter, Mandy, told the London Evening Standard
that she would change her vote if she could.
"This morning the reality is actually hitting in and the regret is hitting in," she said Friday. "I wish I had the opportunity to vote again, simply because I would do things differently."
Meanwhile, Cornwall in southwest England -- a region which voted to leave the EU -- is now seeking confirmation from government ministers that it will keep getting funding equal to its previous EU allocation.
"Now that we know the UK will be leaving the EU we will be taking urgent steps to ensure that the UK Government protects Cornwall's position in any negotiations," said John Pollard, the leader of Cornwall Council.
"We will be insisting that Cornwall receives investment equal to that provided by the EU program which has averaged £60m (more than $82 million) per year over the last ten years."