In-form North End midfielder Barry Nicholson scored a fine opener in the 23rd minute as the hosts looked to record a badly-needed first win since Phil Brown took charge last month.
Yet the Clarets drew level through Jay Rodriguez in the 32nd minute and on-loan Chelsea midfielder Cork sent a diving header into the net to settle the match six minutes from time.
Preston were desperate to build on recent encouraging performances by finally recording a win under Brown.
The former Hull boss had clearly fired his players up for the match and their desire and commitment could not be faulted.
The hosts began brightly and in the fifth minute Billy Jones hit a fierce shot from 30 yards which sailed just wide.
Three minutes later Preston threatened again when striker Nathan Ellington saw a 20-yard shot blocked by Cork.
Burnley were not without chances of their own and in the 10th minute winger Chris Eagles engineered the space to hit a 20-yard shot which on-loan Everton goalkeeper Iain Turner blocked.
Yet the momentum remained largely in Preston's favour, with in-form winger Keith Treacy having a shot blocked by Clarke Carlisle.
Ian Ashbee then missed the target from 25 yards but his side led moments later when Nicholson finished off a fine flowing move.
Ashbee was involved in a superb passing sequence which culminated in Nicholson latching onto a pass from Paul Coutts before sweeping a 12-yard shot high into the net to send Deepdale into raptures.
Burnley boss Eddie Howe boldly started with two forwards as Chris Iwelumo began up front alongside Rodriguez.
The Clarets steadied themselves and hit back in the 32nd minute when Eagles produced a fine pass which Rodriguez latched onto and rifled into the bottom corner from 12 yards.
Yet the response from Preston was impressive and they almost forged back in front in the 36th minute when a cross from Nathan Ellington was blocked for a corner by Michael Duff.
Treacy delivered an inswinging corner and defender Craig Morgan rose to head the ball goalwards only to see his effort blocked by Cork.
Eagles, who was Burnley's most creative player, hit a shot wide from 12 yards and went close from 20 yards as the sides went into the break level at 1-1.
Shortly after the restart, Rodriguez had a shot blocked by David Gray and Coutts then saw his effort blocked by Carlisle at the other end.
Eagles came increasingly to the fore for the visitors and in the 63rd minute his cross was headed wide by Rodriguez.
Nicholson then had a shot parried by Burnley goalkeeper Lee Grant before Eagles tested Turner at the other end.
Both sides pushed for a winner and it was Burnley who got it when Wade Elliott crossed from the right flank and Cork arrived on cue to head home.