The Gunners saw Edward Nketiah sent off following a VAR review, before Vardy levelled with a lengthy delay for possible offside.
Vardy and Kelechi Iheanacho had early chances as Leicester started well, but it was Arsenal who took the lead in the 21st minute thanks to a wonderful piece of play from Bukayo Saka. The England under-19 international was fleet of foot to open up Leicester, wrong-footing Jonny Evans to lay on a simple chance for Pierre-Emerick Aubameyang to score.
Saka then almost added a second himself, but Kasper Schmeichel kept the effort out with his legs. And the 18-year-old was again at the centre of the move as Schmeichel got down low to keep out Alexandre Lacazette.
Nketiah came off the bench for Arsenal in the 71st minute, but within four minutes he was back down the tunnel after being sent off following a VAR review. Nketiah had been booked by referee Chris Kavanagh for a challenge on James Justin, but the VAR Stuart Attwell advised it should be upgraded to a red -- leaving Arsenal to play out the rest of the game with 10 men.
From that point onwards Leicester dominated the play, and got themselves back on level terms in the 85th minute when Demarai Gray's inch-perfect ball was finished off by Vardy from close range. Another lengthy VAR review followed, with Ayoze Perez stretching for the ball in an offside position; touching the ball would have played Vardy offside, but Perez was deemed not to have reached it with his outstretched boot.
Leicester couldn't force a winner, with Dennis Praet firing over in the final seconds, and the two sides had to settle for a point.
"We lost two points. I think we deserved to win the game," Arsenal boss Mikel Arteta told the BBC after the match. "But when you have the chances you have to kill them. Three or four-nil and the game is over. We had to play a long time with 10 men.
"I was agitated for a previous incident that was a red card [in the first half]. The referee has made a decision and we have to learn from that, as well it is our fault because it is the fourth time since I have been here we have played with 10 men and we have to reflect on that.
"Knowing Eddie no, looking at the challenge on the camera arguably it can be a red card or not. He will learn from that because he is such a young kid.
"It is where we want to head. I am really proud because a lot of things we worked on happened, the players are believing in the way we want to play, how aggressive we want to be. Just that little extra quality to finish the actions better.
"The boys played really well credit to them. I am so proud. We take the positives, keep improving and go again against Spurs and try to win."
Arsenal's next Premier League match will be at Tottenham on Sunday, while Leicester head to relegation-threatened Bournemouth.