Jurgen Klopp was without a host of key men on Monday (AEDT), but Liverpool produced a dominant performance and clinched a victory that helped secure a slice of history and take them second in the table.
Liverpool's previous record of 63 matches unbeaten in the league at Anfield was set in December 1980, the run having started more than two years earlier in February 1978.
Leicester was the team to end the run on that occasion, though it was unable to inspire a repeat of history.
Jonny Evans' own goal was added to by Diogo Jota, who himself set a new Anfield record as he became the first Liverpool player in history to score in his first four home league games.
Roberto Firmino then completed the scoring late on with a header.
Liverpool's immense 64-match streak began rather unremarkably on 8 May, 2017 with a 0-0 draw against Southampton, having last lost at home in the Premier League on April 23.
A Crystal Palace team inspired by ex-Reds striker Christian Benteke were the last visitors to prevail at Anfield in the league.
Leicester travelled to Merseyside having won all of their previous five away games in the league this term, but former Liverpool boss Brendan Rodgers could not mastermind what would have undoubtedly been a satisfying feat for the Foxes manager.