Barkley produced a stunning solo run from inside his own half before powering into Newcastle territory and firing into the roof of Tim Krul's net.
Osman picked out Gerard Deulofeu on the right wing shortly after the restart and the Barcelona loanee slid an inch-perfect square ball for Lukaku to crash home.
The visitors added a third shortly before time as Deulofeu and Lukaku linked up to find Everton captain Osman, who struck into the top right-hand corner with power and precision.
It was the hosts who made the brighter start on Tyneside as Luuk de Jong and Yoan Gouffran both had an opportunity to break the deadlock inside two minutes.
Dutch forward De Jong's off-balance shot from the edge of the box was denied by the outstretched knee of John Stones, before Gouffran - who was unchallenged in the area - entirely missed the ball following a return ball from Moussa Sissoko.
Everton's riposte came in the form of a change of pace from Spanish attacker Deulofeu, who skirted down the right flank, beating two defenders, but was unable to find Lukaku in the centre.
Roberto Martinez's side could have taken the lead after 15 minutes, Osman's first-touch pass from Deulofeu's squared ball released Lukaku in the box.
The Belgian striker's initial touch took the ball away from Fabricio Coloccini, but his low shot was well saved by a sprawling Krul in the Newcastle goal.
Everton eventually took the lead midway through the first period as England midfielder Barkley produced a magnificent solo display.
The 20-year-old gathered possession 70 yards out before powering past a host of Newcastle defenders, drifting from right to left as he advanced into the box, where he fired ferociously into the roof of Krul's net.
Newcastle were sharp in response as Vurnon Anita and Papiss Cisse exchanged passes on the edge of the area, before the latter was crowded out by Stones and Seamus Coleman.
Despite holding the advantage, Everton spent a prolonged period supressed in their own half and Cheick Tiote - fresh from striking the woodwork on Saturday - shaped a stinging 25-yard effort toward goal, before a bravely interjection from the head of Sylvain Distin.
Everton's superior pace shone after the break as they doubled their advantage within seven minutes of the restart.
Osman's precision cross-field ball picked out Deulofeu on the right wing and his first touch took him away from Paul Dummett before offering an inch-perfect square ball for Lukaku to thump home his fourth goal in as many matches from three yards out.
Gouffran sought to keep Newcastle in contention as he attempted an audacious first-time volley from Tiote's cross on the edge of the box, only for his effort to soar over the crossbar.
The introduction of Hatem Ben Arfa brought a much-needed injection of creativity and the France international soon jinked between three Everton defenders before squaring to Anita, who was unable to convert from five yards, squandering the hosts' best chance of the second half.
The wrapped up victory with a third goal shortly before full time Deulofeu and Lukaku linked up to find Everton skipper Osman in space on the edge of the box and struck into the top right-hand corner of Krul's net with venom.
A first away win of the calendar year sees Everton elevated to fifth in the Premier League standings, one point ahead of Tottenham, who they still have a game in hand on.