Vardy rounded Clarets goalkeeper Nick Pope from substitute Kelechi Iheanacho's through ball with five minutes remaining to complete an eventful afternoon for the Foxes striker.
Burnley's summer signing Maxwel Cornet had scored a stunning volley for the visitors late in the first half on his first league start since joining for £12.85m from Lyon in August, firing in emphatically from Matej Vydra's cross.
Sean Dyche's side had taken an early lead when Vardy inadvertently beat Kasper Schmeichel with a flicked header at his near post from Ashley Westwood's corner.
However, the Foxes striker atoned for his error as he raced on to Youri Tielemans' pass and found the bottom corner - before rescuing a late point with another calm finish.
Burnley boss Dyche, who was forced to replace the impressive Cornet because of injury in first-half stoppage time, thought his side had scored a dramatic winner in injury time only to see Chris Wood's header ruled out for offside.
Dyche later said: "We lost Jay Rodriguez to a hamstring injury [in training] but I thought Matej Vydra and Chris Wood were excellent. Maxwel Cornet had a good performance but we lost him to injury too."
On Cornet's injury, he added: "We will wait and see, it's too early yet but I think it is his hamstring, yes."
On waiting for a first win, the manager also said: "We have to earn the right. I don't want to sound like a broken record, the performance was decent again. It's that balance - I'm aware of how the team is performing but you have to turn it into wins."
Need a VPN? Or seeking to switch? Then support Tribal Football by clicking here to sign up with our official partners NordVPN. Not only will you receive the best VPN experience, but also great discounts by signing up via Tribal Football. Thanks for your support!