Trippier turned the ball past his own goalkeeper after a mighty goalmouth scramble in the 64th minute which largely summed up an untidy and uninspiring affair.
Not that it will matter to Garry Monk’s buoyant visitors who seized their chance to build on last week’s famous win over Manchester United and maintain their hopes of a top-10 finish.
Bafetimbi Gomis had missed the best chance of the first half but the hosts will count themselves aggrieved to get nothing after just about having the better of a gruelling game.
Ashley Barnes – showing no ill effects from the reaction to his challenge on Nemanja Matic last week – looked bright up front while George Boyd and Michael Kightly also had chances.
The hosts had a strong penalty claim rejected after Federico Fernandez appeared to handle in the box, and Swans keeper Lukasz Fabianski pulled off two fine late saves to deny first Barnes then substitute Sam Vokes.
For long spells it looked like Gomis would be left to rue his awful first-half miss when he sprang the hosts’ offside trap only to splay his effort wide of Tom Heaton’s goal.
Moments earlier Gomis had served up what passed for the visitors’ only shot on target in the opening 45 minutes when he drilled a low shot straight at the Burnley keeper.
But for Swansea’s late rally it was the home side who had the better of a period big on desire and commitment but lacking in quality.
Both sides began surely buoyed by their respective performances against Chelsea and Manchester United last week but it soon became clear there would be few glimpses of a similar level of excitement.
Boyd turned in the box to hit a low shot straight at Swansea keeper Fabianski with four minutes on the clock as the hosts started well.
Intelligent work by Danny Ings in the 23rd minute served up a chance for Barnes, who seemed surprised by the opportunity and slammed his effort straight at Fabianski from point-blank range.
Burnley looked brighter after the break and had a strong penalty shout turned down by referee Jonathan Moss in the 53rd minute.
Kightly rescued a lost causes on the left and cut along the goal-line to hit a shot which Fabianski parried, and the ball appeared to strike the arm of Fernandez before being cleared.
As the hosts began to stretch the game Barnes came close on the hour mark, latching onto a long ball from Trippier and controlling superbly before firing inches wide of Fabianski’s left-hand post.
But just as Burnley looked more likely to break the deadlock it was Swansea who grabbed the lead following a mighty goalmouth melee.
Gomis flicked on a Shelvey cross to Cork, whose initial effort at the far post was brilliantly tipped against the bar by Burnley keeper Heaton.
The rebound again found Cork whose second shot across goal struck the unfortunate Trippier and was deflected into his own net off the goalkeeper.
Burnley hustled gamely for an equaliser with the ever-bright Barnes attempting a snap-shot from the edge of the box which was well saved by a diving Fabianski.
And in the 85th minute the hosts came even closer when Boyd crossed from the right and Vokes’ close-range strike was brilliantly parried by the keeper.
Despite a generous five minutes of stoppage time the hosts did not come up with another chance and their top-flight future looks ever more precarious.