The No 10 seed was pushed all the way by her Belarussian opponent who finally succumbed 6-2 6-7 (5/7) 6-4 after two hours and 40 minutes.
However, it all appeared to be plain sailing for Britain's No 1 when she wrapped up the opening set in just 33 minutes, thanks to two breaks of serve.
In the second, Konta looked in control once again as she moved into a 4-2 lead - and had a point for 5-2 - before the 23-year-old from Minsk staged a brave recovery.
