Roberto Firmino came off the bench to score with his first touch as Liverpool came from behind to maintain their unbeaten Premier League start after Jurgen Klopp's decision to ring the changes almost backfired.
It means Liverpool remain second, cutting Manchester City's advantage at the summit to two points with a 12th victory of the campaign, and in the process posting the club's best start to a top flight season after 15 games.
They were pushed all the way by a Burnley side now anchored in the relegation zone after an eighth game without victory.
A totally different outcome looked possible when Cork pounced from close range after keeper Alisson twice failed to gather the ball after confusion in the Liverpool box from a Johann Berg Gudmundsson corner.
The lead lasted just eight minutes before James Milner levelled with a cool 20-yard finish into the bottom corner from an inviting Divok Origi lay-off.
Klopp immediately threw on Mo Salah and Firmino in a bid to seal victory which was rewarded in the 69th minute when Firmino guided the ball into an unguarded net after Virgil van Dijk redirected a Trent Alexander-Arnold free-kick across the face of goal.
Liverpool's German head coach utilised his squad to make seven changes from the side that clinched a dramatic stoppage-time Merseyside derby victory over Everton.
Skipper Jordan Henderson returned from suspension for his 300th career league appearance as the Reds started a game without any of Salah, Sadio Mane or Firmino for the first time in more than 18 months.
After midfielder Ashley Westwood fired over early on, Burnley had to settle for shots from distance in an attempt to find a way past Alisson, with Chris Wood and Phil Bardsley both going close from long range.
The hosts had the ball in the net as half-time approached, but Ashley Barnes' fine first-time 18-yard volley into the top corner from Robbie Brady's free-kick was correctly ruled offside.
Liverpool started the second-half with greater purpose, and Daniel Sturridge forced a good save from Hart with a powerful effort from the edge of the area which was destined for the bottom corner.
Burnley's former England keeper further excelled when he managed to fingertip a fierce Naby Keita effort onto a post and out for a corner after the Liverpool midfielder burst past Ashley Westwood.
The hosts responded with their best spell of the game as the hour mark approached, and a 300th league game in management for Burnley's Sean Dyche's looked like being a fruitful one after Cork pounced for his third goal of the season.
That was until Milner's leveller changed the complexion of the game, and Klopp quickly called on his cavalry to provide the desired result.
Alisson redeemed himself in stoppage time with a great save from Mee's header before launching the move that led to Liverpool's third goal when Xherdan Shaqiri swept home a Salah-inspired move to leave the Clarets only off the bottom only on goal difference after taking two points from the last possible 24.
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