Nathan Dyer scored in stoppage time to give the Swans the lead at the interval against the in-form Magpies.
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Chicho Flores struck a long-range effort for the home side on 26 minutes, but Tim Krul was able to watch it flash wide of the post.
Remy had another chance on the half-hour mark, but the Frenchman dragged his shot
wide from distance after finding some space between midfield and defence.
With the half seemingly heading for a goalless draw, Dyer struck when he headed home in stoppage time.