Coventry looked to be on the brink of third-tier football for the first time since 1964 but Platt's header, within 11 minutes of stepping off the bench, combined with Birmingham's comeback to salvage a 2-2 draw against relegation rivals Bristol City keeps their hopes alive on Saturday.
The Sky Blues remain four points adrift of the Robins, who occupy the final position of safety, but now have just three games to save their Championship future - with bottom club Doncaster's relegation confirmed on Saturday.
Charlie's Austin's header, his 16th league goal of the season, handed Burnley the lead in the 20th minute but Eddie Howe's side now sit eight points off the top six.
Burnley-born Oliver Norwood, a former season ticket holder at Turf Moor, was first to threaten for Coventry with his early effort easily saved by Lee Grant.
But the Sky Blues went much closer after 10 minutes when Richard Keogh's header from Sammy Clingan's corner was cleared off the line by Kieran Trippier, with the assistant keeping his flag down despite appeals from the visitors.
And from the recycled play, recalled striker Alex Nimely burst past two players before forcing a solid parried save from Grant.
The lively on-loan Manchester City forward tested Grant again soon after as Coventry made a bright start to proceedings.
But it was Burnley who made the breakthrough with their first real attack, Ross Wallace's left-wing cross headed home by Austin.
The goal stunned Coventry, who were previously firmly on top, and the Clarets, far from at their best, were lucky to escape punishment for Wallace's dangerously high challenge on Jordan Clarke.
A lull in the game was broken by Danny Ings' effort in the 35th minute which was well saved by Joe Murphy.
The City stopper was called upon again to deny Austin before the Sky Blues enjoyed a brief period of pressure towards the end of the first half albeit without testing the scoreline.
Cody McDonald saw an effort blocked for Coventry early in the second half.
Junior Stanislas fired off target from the edge of the penalty area at the other end before Dean Marney did the same from a similar position.
Coventry boss Andy Thorn made a triple substitution in the 56th minute in a bid to change his side's fortunes - Platt, Gael Bigirimana and Cyrus Christie all coming on.
And it was Platt who made an almost immediate impact, heading home Norwood's cross from close range for a crucial equaliser.
The Sky Blues suffered a blow with 15 minutes remaining as Christie was forced off due to injury, leaving them with 10 men for the closing stages.
Wallace's long-range effort brought a stop from Murphy, who then thwarted Austin late on to preserve Coventry's point.