England resumed on 56 without loss, chasing a colossal winning target of 628 or needing to bat for two full days for a draw – both outcomes considerably less likely than a hefty defeat at the start of play.
Keaton Jennings' dismissal for 17 was frustratingly familiar, caught at slip off a seam bowler, but Burns made an assertive 84 before being clean bowled by an innocuous delivery from spinner Roston Chase.
That left the tourists 134 for two and in need of a major rearguard action from Jonny Bairstow (29 not out) and the middle order.
Burns, 39no after a positive start the previous evening, saw his second scoring shot zip wide of the slips for four but also dispatched a couple of early half-volleys to settle in.
The Surrey skipper reached his second Test fifty in 79 balls, acknowledging with a low-key wave of the bat, and will have been just as pleased to pass his previous best of 59.
While his innings was progressing nicely, Jennings had slowed to a crawl by the time Alzarri Joseph slanted one across him and drew the error. Given the circumstances the decision to flash outside off was as regrettable as the edge which flew swiftly in the direction of Jason Holder at second slip.
Jennings has now scored 20 or fewer in 19 of his 30 Test innings and looks to be clinging to his place. Bairstow proved a livelier foil for Burns, reeling off a handful of boundaries including three in an over from England's first-innings conqueror Kemar Roach.
When Chase was introduced just before lunch Bairstow muscled his first delivery wide of mid-on but moments later Burns left a gap between bat and pad and was castled by a straight one.