Meanwhile, Spurs were top of the table before their midweek clash with Liverpool but now find themselves six points adrift of the champions after a damaging 2-0 defeat at home.
Manchester United demolished Leeds 6-2 in the first league meeting between the sides for 16 years to bolster their Premier League title aspirations.
Scott McTominay and Bruno Fernandes scored twice in a match on Sunday as United's counter-attacking threat tore apart Marcelo Bielsa's attacking approach to move the Red Devils up to third, five points behind leaders Liverpool with a game in hand.
Earlier, Leicester put a major dent in Tottenham's Premier League title challenge, climbing to second in the table as Manchester United prepared to resume their historic rivalry with Leeds.
Vardy put away-day specialists Leicester ahead with the final kick of the first half after Brendan Rodgers's side were awarded a penalty.
Serge Aurier barged Wesley Fofana over and, after referee Craig Pawson watched the incident again on the pitchside monitor, the Foxes striker drilled home his spot-kick.
Spurs had a huge escape early in the second half as James Maddison had a goal ruled out for the most marginal of offsides.
But Leicester doubled their lead just before the hour mark via an own goal.
Vardy capitalised on some bad defending by Moussa Sissoko and his header across goal deflected off Toby Alderweireld and into the net.
The win, Leicester's sixth in seven away games, lifts them to within four points of Jurgen Klopp's men after 14 matches and will have Foxes fans dreaming again.
Just five points separate the 2016 champions from Chelsea in eighth spot but Liverpool looked ominous in a 7-0 demolition of Crystal Palace on Saturday.
In the early kick-off on Sunday, Sheffield United were cruelly denied their first Premier League win of the season as Danny Welbeck scored a late equaliser for Brighton in a 1-1 draw.
Chris Wilder's bottom-placed team looked set to claim all three points at the Amex Stadium despite being reduced to 10 men late in the first half.
Substitute Jayden Bogle marked his Premier League debut with the opening goal for Sheffield United in the 63rd minute.
But Welbeck's goal left United with just two points from 14 games, eight points from safety.