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The South African fired down 14 aces as he eased past the Lithuanian by a 6-2 6-2 7-6 scoreline.
Anderson required over three hours to see off Diego Schwartzman in the first round, but Wednesday's match proved to be a much easier affair.
The 14th seed dropped just one point on serve as he wrapped up the first set inside 23 minutes, before another two breaks in the second helped him into a commanding lead.
Berankis offered more quality in the third as he attempted to remain in the match but despite managing to take the set to a tie-break, he failed to derail his opponent who won in one hour 47 minutes.