Bony kept his cool from the spot in the first minute of time added on at St James' Park to move the Welsh side six points clear of the drop zone with three games remaining after substitute Marvin Emnes had been brought down by Cheikh Tiote.
The Ivorian had earlier equalised with a powerful header in first-half stoppage time after Shola Ameobi's first top-flight goal since December 2012 put the home side in front.
Swansea coach Garry Monk described a reported training ground bust-up as "handbags" this week, and was more interested in ensuring his side get out of "a serious situation" at the wrong end of the table.
And Monk's team showed no shortage of unity on Saturday as they secured their first away win in the Premier League since November to surely end any fears of returning to the Championship.
Defeat for Newcastle was their fifth in a row and heaped more pressure on manager Alan Pardew, who watched from the stands in his final game of a seven-match touchline ban for headbutting Hull City's David Meyler.
There was drama in the first minute as the ball struck referee Chris Foy in the face, although the official was initially able to continue after the being assessed by the Newcastle physio.
Newcastle fans soon became restless as Swansea got the upper hand and Pardew was forced into a change only 21 minutes in, when Papiss Cisse picked up an injury and fit-again full-back Mathieu Debuchy replaced him.
Pardew's side had looked devoid of ideas, but they were in front out of the blue after 23 minutes.
Recalled striker Luuk de Jong took advantage of some poor Swansea defending as he turned in the penalty area and picked out Ameobi, who got the ball out of his feet and finished past Michel Vorm to end Newcastle's goal four-game goal drought.
Fourth official Anthony Taylor then replaced Foy on the half-hour mark as a result of his early blow to the face.
Newcastle were forced to make another change five minutes before the break, with leading scorer Loic Remy making his eagerly-awaited return from injury in place of De Jong after the Dutchman landed awkwardly on his ankle.
The home side looked set for a half-time advantage, but Bony had other ideas as he rose highest to head home Ben Davies' inviting corner a minute into time added on.
Newcastle almost restored their advantage just after the break, but Swansea keeper Vorm got down to his left to keep out Dan Gosling's strike.
Vurnon Anita then flashed a left-foot strike just wide of the far post, but both sides lacked quality in the final third in a frantic second half.
Swansea's Jonathan de Guzman came close 12 minutes from time, but his right-foot strike found the roof of the net.
Bony scented a winner when he raced clear eight minutes from time only to be thwarted by Tim Krul, but he was not to be denied from 12 yards as he sent the Dutchman the wrong way after Tiote had upended Emnes.
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