Matthew Tait scored Newcastle’s try and Burke provided the other points with his unerring accuracy to leave the Falcons looking towards mid-table after back-to-back wins in the league.
Tykes remained bottom after fly-half Gordon Ross missed chances to build on a try from Andre Snyman.
Leeds’ second biggest attendance in their history, of 11,601, ensured a 10-minute delay for congestion, but fans were treated to a match which could have gone either way once play started.
After a four-week hiatus for cup rugby, it was the hosts who started brighter, although they failed to convert their early possession into points.
Tom Biggs twice came close, firstly by creating room on the left wing with a grubber kick that James Grindal had to kick into touch a metre from the line.
Biggs attacked the other wing after 12 minutes and was only denied a try by Jamie Noon’s tackle, with the Falcons centre receiving treatment from the physio for his troubles.
The visitors repelled Leeds’ early pressure and burst into life through the pace of Tait.
The 19-year-old Tait – sporting distinctive purple cycling shorts – took advantage of Andy Long’s surging run in the 15th minute, only to be stopped by Tom Palmer when a try looked certain.
He was not made to wait long, with the first points of the day coming from the next phase of play. Toby Flood’s delicate kick sent Tait through to beat Biggs to the ball. Burke converted.
Ross missed an opportunity to reduce arrears with a penalty from the restart, but then got the hosts on the board with a kick in the 24th minute.
Burke restored the seven-point advantage with a penalty from the halfway line before Leeds launched a period of sustained pressure.
Justin Marshall was the architect of Leeds’ opening try, offloading to Andy Perry in the 38th minute five metres out with an improvised pass over his shoulder. Perry supplied Snyman and the South African wing scampered over. Ross added the extras to level the scores at 10-10.
Burke’s impressive kick from the left touchline ensured a slender half-time
lead for the Falcons.
The second period started with a succession of wayward kicks and infringements, and only a magnificent run from Cory Harris in the 50th minute brightened proceedings.
Ross then missed the chance to level the scores with a penalty that fell short, although the kick was from the halfway line. The disappointment was compounded by a Leeds try being ruled out by referee Chris White for a forward pass in the 55th minute.
Falcons had brought on Dave Walder on for Flood at the interval but his creativity could find no way past Leeds, who looked to mount a late challenge.
Leeds threw everything forward in the last moments but Falcons held on for a priceless win.