The Magpies had taken a first-half lead, but missed chances in the second period as Bony goals at the end of each half secured the win for the away side.
After a bright start from Swansea in which Pablo Hernandez looked lively, Newcastle improved and took the lead after 23 minutes when Shola Ameobi managed to bundle his way into the box before firing the ball into the bottom corner.
Swansea increased the tempo after that but neither side really tested the opposition goalkeeper until first-half stoppage time, when Bony found space in the box and headed home Ben Davies's cross from close range.
Swansea managed to turn the tide back in their favour and created a couple of chances in quick succession as the half wore on. First, Bony headed straight at Tim Krul from Jonjo Shelvey's lofted pass, before Hernandez curled just wide from the edge of the box.
Jonathan de Guzman then attempted a similar effort from just outside the box and went even closer, clearing the crossbar by inches.
Swansea created an even better chance five minutes later when Bony powered his way into the box and around Krul, only for the Dutch goalkeeper to get back and get something on the ball to clear the danger.