Mathew Tait scored Newcastle's try in the first half 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 remain bottom after fly-half Gordon Ross missed chances to build on a try from Andre Snyman just before the interval.
Barry Everitt salvaged a dramatic 20-19 victory for London Irish over Saracens with the last kick of a hot-tempered game at Vicarage Road.
The fly-half fittingly drilled over a 40-metre penalty to finish with 15 points, having also supplied the creative spark for Michael Horak's first-half try.
While Everitt was booting his side into the Premiership's top four, Saracens were left kicking themselves after an error-strewn game they took an hour to get to grips with.
Glen Jackson appeared to have dug them out of a hole a minute from time when he landed his fourth penalty to go with his conversion of Hamish Mitchell's late try.
But ultimately his side paid for failing to take some glaring chances and for Jackson's inexplicable misses from close range before the break.
Jason Strange punished a poor Leicester performance by kicking Bristol to an emphatic 15-3 victory at the Memorial Stadium.
Strange booted five first-half penalties as Bristol celebrated a record home league crowd of just under 12,000 by recording their fifth Premiership win since regaining top-flight status this season.
They should have won even more convincingly against an under-strength Tigers outfit and, if their finishing had matched Strange's accuracy, Leicester could have suffered an embarrassing loss.
The Tigers' only points came from a Ross Broadfoot penalty in the second half.