Osaka, the former No 1-ranked played who won the U.S. Open two years ago, played far cleaner tennis than her opponent in a 6-3, 6-4 win over 93rd-ranked Shelby Rogers.
Rogers finished with 27 unforced errors, Osaka with eight.
Earlier, Brady, the 28th seed, defeat Yulia Putintseva of Kazakhstan 6-3, 6-2.
Brady has never been this far at a major tournament but has split her two previous matches against Osaka, winning in 2014 before the Japanese star squared the ledger four years later.