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Williams, who won the Wimbledon title on her last two visits in 2015 and 2016, is now on an 18-match winning streak at the All England Club.
The 21-year-old Latvian former French Open champion looked in trouble with her Belarusian opponent racing into a 5-2 lead after delivering a superb lob.
The women's top seed Simona Halep crashed out in the third round of Wimbledon on Saturday when she was beaten 3-6, 6-4, 7-5 by Taiwan's Hsieh Su-wei.
Marin Cilic, the 2017 runner-up, suffered a shock 3-6, 1-6, 6-4, 7-6(3), 7-5 defeat by Argentine Guido Pella.