The Clarets thought they had equalised six minutes from time when Jay Rodriguez scored, but the offside flag was raised denying the visitors a share of the spoils.
Story of the game
Despite Federico Fernandez impressing after Ciaran Clark's dismissal in midweek, The Magpies boss opted to bring back captain Jamaal Lascelles instead of promoting the Argentine to the starting lineup.
Ryan Fraser was taken off after 12 minutes on Tuesday in place of defensive reinforcements, but injured himself in training during the week meaning it was a first start since early November for Miguel Almiron.
It was a cagey opening to the tie, but Burnley were the side who settled the better, beating Newcastle to the punch almost every single time.
It took 13 minutes for the visitors to have the first big chance of the game when the ball fell wide to Johann Berg Gudmundsson, who had a deflected effort from just outside of the penalty area cannon off the post.
The warning signs were there early on for this winless Newcastle side, but it seemed the pendulum swung in their favour after Burnley's star man Cornet was forced to come off with a thigh injury.
And 10 minutes after that, the hosts broke the deadlock.
Callum Wilson then picked up the ball and took it round the rooted goalkeeper before knocking it into top corner to give his side the lead just half-time.
Burnley did have a big chance before the interval to equalise when Matthew Lowton played a cross-pitch pass into the path of an unmarked Matej Vydra, but he lashed his effort over-the-crossbar to ensure the hosts went into the break with the advantage.
Toon hang on to claim precious three points
Looking to build on their lead, Newcastle came out of the blocks for the second half the quicker of the two sides.
Allan Saint-Maximin was nullified relatively well by Burnley in the opening 45 minutes, but he looked lively shortly after the restart when he forced Pope into a save after he worked space on the edge of the area before shooting.
Pope was forced into action five minutes after that chance when Wilson played through Almiron, who managed to shoot low and right of the England international's goal, but he tipped the effort round the post.
It took another 15 minutes for another chance to arise, and once again it fell to the hosts.
This time Saint-Maximin drove down the left-hand side and put a low cross in towards Joelinton, but James Tarkowski got an important toe on the ball and managed to hook it away.
Tarkowski proved his worth to Burnley again a few minutes later when he his body in front of Almiron's shot.
The Clarets were hanging in 10 minutes from time, and they made two vital blocks to keep themselves in the match.
First off, Almiron's curling strike – that looked destined for the top corner – was blocked heroically by defender Nathan Collins, before Pope made a brilliant save to deny Shelvey's effort from around 25 yards out minutes later.
It looked like Burnley's defending had paid its dividends on 84 minutes when Jay Rodriguez latched onto a ball from Vydra and found the back of the net, but the offside flag was raised to ensure Newcastle kept their lead in tact and bagged their first win of the season.
Newcastle: Dubravka; Manquillo, Lascelles, Schar, Lewis; Shelvey, Willock; Almiron, Joelinton, Saint-Maximin; Wilson. (4-2-3-1)
Burnley: Pope; Lowton, Collins, Tarkowski, Taylor; Gudmundsson, Westwood, Brownhill, McNeil; Cornet, Wood. (4-4-2)