Bialkowski, in the process, kept alive the Magpies' slim chances of reaching the play-offs as the Robins failed to find a way past the former Southampton stopper.
Bialkowski saved his best save for the dying minutes when it seemed the host's pressure must finally pay off.
Adam Rooney headed against a post in the 76th minute but then Bialkowski twice used his long legs to deny Alan McCormack and then in the last minute James Collins.
Earlier he had kept out efforts from Louis Thompson, Tommy Miller and Simon Ferry as County clung on grimly.
Chris Kiwomya's team did have their chances on the break though, with Swindon 'keeper Wes Foderingham saving from captain Neal Bishop and fellow midfielder Alan Judge.
And Enoch Showunmi saw his effort from a corner deflected on to the crossbar by Robins forward Andy Williams.