Magpies skipper Jamaal Lascelles - who had earlier produced an exceptional goal-line clearance to deny Tammy Abraham - headed the victor from Matt Ritchie's 76th-minute corner to end Newcastle's run of 12 matches without a Premier League away win.
Newcastle, in contrast, executed their game-plan perfectly, aided throughout by Benítez's open telephone line to his backroom staff.
The match was nothing like as enjoyable for Paul Clement.
They nearly did go in front when Ayoze Perez forced Fabianski into saving with his outstretched leg.
"I had a good feeling going into the game with the atmosphere around the squad - then there's the actual game and what I saw was not what I expected". We got the overload in midfield but we never got control of the game.
Paul Clement: 'Swansea City were not good enough'
"I'm really not satisfied with the way we played". We are taking everything as it comes at the moment, getting the training right and make sure we are at it again. We will be hard to beat back home but if we play like this every week, we will do well. Swansea may have one more point than Newcastle but Newcastle have a better goal difference and have scored more goals. One-on-one with Rob Elliot, the striker rounded the Newcastle goalkeeper but looked on in despair as Lascelles, covering brilliantly, made a sliding intervention to clear the ball behind. "Hopefully he gets well soon and gets back with us", he said.
"With Matt Ritchie's delivery, it is on a plate every time. It is a performance I am not proud of", manager Paul Clement said.
A win over Swansea could further help boost confidence, but a loss would only complicate matters for Newcastle and its management.
A surprise deadline-day loan signing from Bayern Munich, Sanches struggled to gel with his new team-mates and was withdrawn after 69 minutes.
"We are very happy with the performance and so is Rafa. But, at the same time, we've got to wake up quickly".