County took the lead in the 35th minute when John Cofie played through Alan Judge and the midfielder took the ball down with his chest before planting a perfectly weighted lob over Carlisle keeper Mark Gillespie.
The visitors almost responded after 42 minutes when James Berrett's shot towards goal was cleared off the line by the Magpies' Alan Sheehan.
Carlisle went close again in the 57th minute when Rory Loy headed towards the top corner but Bialkowski clawed the ball away superbly.
The away side then looked to have levelled in the 60th minute when Sean O'Hanlon shaped to tap-in at the far post but somehow Bialkowski came out with the ball.
Bialkowski was the hero again when he brilliantly saved a Berrett shot low down, before Carlisle skipper Liam Miller was sent off in the 89th minute for a headbutt on Joss Labadie.