The 19-year-old netted his first away goal of the season to push the Black Cats into third, five points behind first place Sheffield United.
Swansea's Zan Vipotnik opened the scoring in the fifth minute with a well taken volley after the ball was headed high up.
The Welsh side then doubled their lead via Liam Cullen from a free kick.
A shot curling away from Anthony Patterson was almost unreachable as it smashed in after hitting the bottom of the crossbar.
The visitors got one back before the half-time whistle thanks to Daniel Ballard.
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Another free kick, this time crossed in by Trai Hume to connect with the diving head of Ballard.
Then, in the 73rd minute, Bellingham made a smart tackle and pass into Dan Neil who drove through the middle of the pitch before letting off a screamer to beat Lawrence Vigouroux.
Bellingham then turned scorer as Chris Rigg sent in a first time low cross into the running midfielder, who beat two Swansea players to the ball.
It sparked wild scenes in the away end as Sunderland took a huge stride towards their overall aim of promotion to the Premier League.
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Le Bris, who switched Bellingham's role within the side to play deeper, has been full of praise for the 6ft 3ins midfielder.
Speaking to Sky Sports earlier this season, Le Bris said: "He's still a young player with the ability to play in many different roles," says the Sunderland boss.
"But I like him as a No 8, because I think he's an offensive midfielder. He can express his power, his ability to run and his ability to press - to link defence and attack."
Jobe, of course, is the brother of Real Madrid superstar Jude Bellingham.
The England sensation has been in excellent recent form, netting five goals and registering four assists in 12 LaLiga games.