Coventry's chances of Championship survival remain alive after they secured a point against a Burnley side whose play-off hopes are all but over.
Charlie Austin gave the home side the lead when he headed in a cross from Ross Wallace.
But Platt nodded in from close range from Oliver Norwood's cross to leave his side four points adrift of safety with three games to go.
Burnley's failure to win means they are eight points from the play-off zone.
Coventry had to play the last 15 minutes with 10 men after Cyrus Christie was injured, with the visitors having already made their three substitutions.
Wallace, who had earlier escaped punishment for a high challenge on Jordan Clarke, went close with a long range strike for Burnley but Sky Blues keeper Joe Murphy was equal to the effort.
Murphy also denied Austin late on as he helped Coventry earn a share of the spoils and leave them still in with a chance of avoiding relegation to the third tier of the English game for the first time since 1964.
The Sky Blues are four points behind Bristol City, who drew at Birmingham.