There were three goals in just 13 minutes to spark drama at the end of an intriguing contest at the King Power Stadium.
After Mason Greenwood and Youri Tielemans had earlier traded brilliant first-half goals, Caglar Soyuncu put Leicester in front before substitute Marcus Rashford hit back on his return from shoulder surgery.
Leicester started proceedings in confident mood, with their high intensity and pressing causing problems for United - and particularly the returning Harry Maguire - trying to play out from the back.
Greenwood fired straight at Kasper Schmeichel and Paul Pogba headed over from a corner between bouts of early Foxes pressure, before United did break the deadlock.
It was a tremendous rocket of a strike from Greenwood, who cut inside from the right flank after some patient build-up play and got past Timothy Castagne before unleashing an unstoppable effort into the top corner with Schmeichel helpless.
Vardy and Kelechi Iheanacho tried to hit back quickly, while Ricardo Pereira came close to an equaliser when his cushioned pass back across goal almost squeezed in at the back post.
Schmeichel made a solid save to deny Cristiano Ronaldo and Pogba shot well wide before Leicester levelled a lively contest with another stellar goal.
Maguire was caught out by Iheanacho, who cut back to Tielemans for an audacious, first-time strike that looped up above David de Gea and in.
The impressive Iheanacho flashed a low strike narrowly wide of the far post and Ronaldo had a rare miskick and then failed to pull off an acrobatic overhead kick as Vardy also touched just wide on the stroke of half-time.
The second period started in lively fashion with opportunities for both Iheanacho and Greenwood, who twice shot wide after being allowed to open up on his left foot.
James Maddison and Tielemans both struck over for Leicester, while Nemanja Matic came close to adding another remarkable goal as his left-footed half-volley whistled narrowly over with Schmeichel at full stretch.
A combination of Schmeichel and the offside flag thwarted Greenwood before Rashford was sent on for his first appearance of the season.
Fernandes and Tielemans both tried their luck again, with De Gea then making a fine save as he pushed the Belgian midfielder’s latest attempt onto the post and wide.
Leicester did get themselves in front when centre-back Soyuncu scrambled home after Daka’s low shot following a corner had deflected off Ayoze Perez.
However, Rashford thundered an equaliser past Schmeichel before straightaway Vardy was teed up by Perez to restore Leicester’s late lead in style.
The late scoring was rounded off in stoppage time, when Daka bundled home at the back post to become the first Zambian ever to notch in the Premier League and ask more questions of Solskjaer.