With the match seemingly heading for a goalless draw, Vokes hooked home from close range off a headed knockdown from Michael Keane to secure Burnley's fifth successive home win.
The ball hit the Wales striker's hand before he scored but referee Mike Dean dismissed Leicester's appeals.
"It was a handball. It was for everybody, for the television, but not for the referee," Leicester manager Claudio Ranieri told the BBC.
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"The referee is the boss. He didn't see it and we lost - in this moment everything is wrong against us, including the decision of the referee."
Burnley have taken all but one of their 29 points at Turf Moor this season while Leicester, with 21 points from 23 games, have yet to win away from home in the current campaign and are now perilously close to the drop zone.
The Foxes are 16th, level on points with Swansea below them and just two clear of the relegation places.
They now have the worst record after 23 games of any reigning top-flight champions and Tuesday's defeat was the first time they had suffered three successive Premier League losses under Ranieri.
Leicester have not won away since they beat Sunderland 2-0 at the Stadium of Light in April 2016, when they were well on their way to a fairytale title.
(Reporting by Alan Baldwin, editing by Brian Homewood, Neville Dalton)
