The Falcons dominated much of the game but frequently failed to make the most of their ample chances until the decisive late try.
It was a much-needed victory for Rob Andrews' men who had failed to win in any of their last five Premiership outings.
Leeds opened the scoring after seven minutes when the Falcons were caught offside at the ruck. Former Newcastle full-back Tim Stimpson kicked the penalty from 25 metres.
Newcastle responded on 17 minutes after a period of pressure on the Leeds line. The Falcons claimed the line-out and the ball was worked left down their back line.
Tom May made the extra man and fed Burke, who powered over in the corner.
The Australian international failed to convert his own try.
Leeds were reduced to 14 men on 25 minutes when winger David Rees was harshly sin-binned by referee Steve Leyshon for entering a ruck from the side.
The Tykes were further hampered four minutes later when England international winger Iain Balshaw was forced off after a nasty collision in the midfield.
Stimpson edged Leeds in front with a second penalty after the home side were again caught offside at the breakdown.
Replacement winger Diego Albanese pushed Leeds further ahead when he finished off some fine work by Rees.
Rees intercepted and made 30 metres down the right touchline, beating three men in the process.
The ball was recycled and fly-half Gareth Wright made a further incision deep into the Newcastle 22 before working the ball left for the Argentinian international to slide over in the corner.
Stimpson's conversion made the score 13-5 to the visitors at the break.
Newcastle, however, started the second half with far more purpose and Michael Stephenson was unlucky not to score after a clever crossfield kick by Dave Walder.
The Falcons continued to spurn chances until Burke scored his second try of the day after Phil Dowson had made 20 metres from a quick-tap penalty.
Walder's conversion made the score 13-12 to the visitors.
Stimpson missed with a relatively short penalty attempt two minutes later but Leeds continued to press and were rewarded with Stimpson's third penalty of the day.
But then Falcons centre Jamie Noon struck on 74 minutes after a searing break from 18-year-old replacement Matthew Tait.
The ball was worked to Noon in the midfield and the England hopeful ducked under a challenge from Chris Murphy before turning on the gas and scorching home under the posts.
Walder's conversion made it 19-16.
Tait was involved again three minutes later when his clever inside pass set up good field position.
Leeds infringed and Walder made it 22-16.
Burke's third try deep into injury time, converted by Walder, provided Newcastle with the flattering scoreline.
For Newcastle:Tries: Burke 3, NoonCons: Walder 2Pen: Walder
For Leeds:Try: AlbaneseCon: StimpsonPens: Stimpson 3