Serena won 7-6 7-5 against Kiki Bertens before Venus won 6-3 6-4 against Sorana Cirstea to tee up a third-round clash.
That will be their earliest meeting at a tournament since the first match they ever played – a second-round tie at the 1998 Australian Open.
Serena’s first-round win earlier this week was her first victory since a 14-month absence after giving birth. She leads her elder sister 17-11 in all-time meetings.
Elsewhere, US Open finalist Madison Keys was eliminated by wildcard Danielle Collins, the world No 116.