The Magpies' 23rd victory of the season in all competitions was a first in six games and Eddie Howe's side had to fight until the last kick at St. James' Park against one of the league's form teams.
Nick Pope made an important save from Jack Hinshelwood inside the first two minutes - then Carlos Baleba's long range curler struck the top of the Leazes End crossbar as Brighton made the stronger start.
But the home side came close themselves in the 11th minute when Osula drifted in from the left and fired just wide.
Seconds later, Jacob Murphy beat Bart Verbruggen to a Bruno Guimarães ball over the top. The Brighton keeper raced from his goal to make a clumsy challenge outside his area but Murphy quickly regained his footing and crossed for Osula to head into the unguarded net for a third goal in his last four matches.
And the Magpies doubled their head in the 24th minute when Burn marked his 150th Premier League appearance by heading home Guimarães' left-wing corner against his former club.
Soon afterwards, a Pope clearance ricocheted off Hinshelwood and against a post but at the other end, good combination between Osula and Willock saw the latter's poked effort kept out by Verbruggen.
It could have been 3-0 when Sven Botman headed Lewis Miley's ball back across goal for Malick Thiaw, but Verbruggen denied the German defender from close range.
A Danny Welbeck header dropped over the bar for the visitors but United had a commanding lead at the break, with Osula shooting wide in stoppage time after a lovely turn and strong run from his own half.
Osula continued to pose a threat for Albion after the break and he was the target when Guimarães raced onto Joe Willock's flick and hit a first-time low cross from the right. Jan Paul van Hecke intercepted, but Verbruggen had to adjust quickly to keep the ball out.
Pope showed similarly impressive reactions to parry a close-range effort from Kaoru Mitoma, although the ball appeared to come off the Japan international's arm. Next, Hinshelwood was off target with a header, then Willock drove wide at the other end.
But - after Yankuba Minteh had lofted wide - Hinshelwood pulled one back. After a sustained spell of Seagulls possession, the England under-21 man drifted into the box, controlled Welbeck's pass and finished superbly.
Newcastle have dropped too many points from winning positions this season but they did just enough to hold on here thanks to Pope's late save from Charalampos Kostoulas' bicycle kick.
Former Magpie Minteh could and should have levelled in the dying seconds when he was presented with a golden opportunity but blazed over from six yards out, and there was still time for Yoane Wissa to shoot off target from Lewis Hall's low centre, and Joelinton to drag wide deep into stoppage time.
But then Wissa teed up Barnes to smash home with virtually the last kick to seal the victory.
Newcastle United: Nick Pope, Sven Botman, Joelinton, Sandro Tonali, Malick Thiaw, William Osula (Yoane Wissa 68), Jacob Murphy (Harvey Barnes 68), Joe Willock (Lewis Hall 68), Dan Burn, Bruno Guimarães (c) (Jacob Ramsey 90, Lewis Miley
Subs not used: Aaron Ramsdale, Kieran Trippier, Anthony Gordon, Anthony Elanga, Nick Woltemade
