Top seed Djokovic enjoyed a 6-4, 6-4, 6-4 win over Germany's Philipp Kohlschreiber with crucial breaks in the ninth game of each set and goes on to face either 2004 champion Lleyton Hewitt or Jarkko Nieminen for a place in the last 32.
"This is the cradle of our sport," said Djokovic, playing his first match since his defeat to Stan Wawrinka at the final of the French Open three weeks ago ended his hopes of completing a career Grand Slam.
Djokovic, also the 2011 champion, hit 12 aces and 36 winners past world number 33 Kohlschreiber.
Top seed Williams, bidding to become the first woman since Steffi Graf in 1988 to wrap up all four majors in one season, endured a nervy start, dropping serve and being warned for swearing, before completing a 6-4, 6-1 win over Margarita Gasparyan, the world number 113 from Russia who has never won a tour match in four years as a professional.
In other early action Monday, Spanish ninth seed Carla Suarez Navarro went down 6-2, 6-0 to Latvia's Jelena Ostapenko in just 52 minutes while Italian 24th seed Flavia Pennetta lost 6-3, 2-6, 6-4 to Kazakhstan's Zarina Diyas.
Two-time semi-finalist Victoria Azarenka, seeded 23, saw off Estonia's Anett Kontaveit 6-2, 6-1 and seventh seed Ana Ivanovic breezed past China's Fan Xu-yi 6-1, 6-1.