Richarlison's early opener and Gylfi Sigurdsson's late penalty gave the Toffees a first win on any ground over Liverpool in 24 attempts dating back to 2010.
This follows the murder of constable Mfihloyakhe Ndlovu, who was killed in the line of duty while protecting a gender-based violence victim in Welbedacht East.
Osaka edged a tight first set and broke the American twice in the second to win 6-4, 6-3 in 77 minutes in front of thousands of fans at Rod Laver Arena.
