Magpies skipper Jamaal Lascelles was the hero at both ends of the field for the visitors, making a goal-saving last-ditch clearance in the second half before soaring highest in the area to score the winner with less than 15 minutes remaining.
The result sees Newcastle end a 12-match winless run away from home in the Premier League as they made it back-to-back top-flight victories for the first time in 21 months.
Swansea, meanwhile, have now lost consecutive Premier League home games for the first time in 2017 and the first time ever in Paul Clement's managerial career.
The match got off to a lively start with Isaac Hayden testing Lukasz Fabianski inside the opening five minutes, but it soon fizzled out as both teams struggled to stamp their authority on the contest.
The Magpies began to apply some sustained pressure for the first time in the match after those two efforts, and it took a stunning stop from Fabianski to keep the scores level when Joselu's goalbound glancing header was tipped wide by the Swansea keeper.
Swansea soon hit back with a good spell of their own, though, and Jordan Ayew forced the first save from Rob Elliot before Alfie Mawson sent a gilt-edge chance wide from the resulting corner when he missed the target with a close-range header.
The early stages of the second half brought with it another talking point as Matt Ritchie was only shown a yellow card for an incident almost identical to the one which saw Sadio Mane sent off for Liverpool against Manchester City on Saturday, with the Newcastle winger having gone in with a high boot on Mawson.
The hosts began to apply pressure again soon afterwards and Tom Carroll was denied by Elliot at the near post before Lascelles made the first of his match-winning contributions with a crucial last-ditch clearance to prevent a certain goal.
Tammy Abraham beat the offside trap to race through on goal and calmly took the ball around Elliot, but Lascelles refused to give up on it and slid in to stop Abraham's subsequent finish from rolling over the line.
Newcastle's own attacking threat started to grow again in the final 20 minutes, with Ayoze Perez firing one good chance straight at Fabianski before the Swansea keeper denied Joselu twice in quick succession.
There was nothing Fabianski could do when the Magpies finally broke the deadlock with a little under 15 minutes remaining, though, as Lascelles rose above Mawson in the box to plant a firm header home from a corner.
Swansea pushed for an equaliser in the closing stages, but Leroy Fer squandered another half-chance from a corner before Luciano Narsingh saw his stoppage-time shot from a tight angle kept out at the near post as Newcastle held on for all three points.
The result leaves Swansea 15th in the early Premier League standings with four points from their opening four games, whereas Newcastle climb into the top half of the table.