After falling behind 2-0 at the start, Azarenka reeled off six straight games and ultimately made quick work of Lapko on a windy day in the California desert.
Konta's pinpoint serving proved too much for France's Pauline Parmentier to handle as the Briton coasted to a 6-2 6-3 win to set up a second-round clash with Hsieh Su-wei of Taiwan.
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Two young Americans also advanced on Wednesday as 17-year-old Amanda Anisimova crushed Serbia's Aleksandra Krunic 6-0 6-4 and 20-year-old Sofia Kenin came from behind to dispatch China's Wang Yafan 1-6 7-5 6-4.
Canadian Bianca Andreescu, fresh off her semi-final appearance at the Mexican Open in Acapulco last week, defeated Irina-Camelia Begu of Romania 6-7(3) 6-3 6-3 to set up a second round battle with Dominika Cibulkova.
Czech Barbora Strycova, China's Zhang Shuai, American Jennifer Brady and German Tatjana Maria were among the other players to advance at the event in Indian Wells.
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