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.
Tammy Abraham came close to scoring after he rounded Rob Elliot, but his shot was superbly cleared off the line by Lascelles.
"After the game Rafa rang and he has congratulated everybody, especially the players because they have done a massive job". There was expectation going in to the game but we were second best in combats, duels, first and second balls, tactics, in all aspects.
They restricted Swansea to few goalscoring opportunities and would have taken the lead earlier but for Lukasz Fabianski producing a wonderful fingertip save to keep out Joselu's header.
Alfie Mawson headed wide for the hosts before the break while Tom Carroll's left-footed volley was well saved by Elliot shortly before Lascelles's saving clearance.
Paul Clement introduced returning striker Wilfried Bony in his place but the fan favourite could do little about Lascelles' thumping victor, the captain getting above Alfie Mawson to leave Fabianski with no chance. It's not one I'm very proud of and we've got to move forward quickly and turn that around.
"We have set a standard now".
Assistant manager Francisco de Miguel Moreno said Benitez had been in contact during the match as his team recorded successive top-flight wins for the first time since December 2015.
The defeat means that Clement has now lost back-to-back home league games for the first time in his managerial career. I think with the players they've got they will be doing well to stay out of the relegation fight.
Mexico: Death Toll In Earthquake Rises To 60
Residents of the Mexican capital fled into the streets, many in their pajamas, for fear buildings would collapse. The worst-hit city was Juchitan, on the narrow waist of Oaxaca known as the Isthmus, where 36 quake victims died.
Sadio Mane has issued a heartfelt apology to Ederson
Liverpool will play Sevilla in the Champions League on Wednesday as City will play away to Feyenoord on same day. Ederson , who took a kick to the face, was replaced by Claudio Bravo, while Mane is now facing a three match ban.
Reaction to draw at Stoke
Stoke skipper Ryan Shawcross will miss Saturday's home game against former club Manchester United due to a back injury. Stoke have failed to beat United in each of their last three head-to-heads and could well come unstuck as a result .