British No. 2 Evans defeated Moldova's Radu Albot in three sets, 7-6 (7-3), 6-1, 6-4.
Ruud, who reached the final at both the French Open and the US Open this year, lost to Jaume Munar 6-3, 6-3.
The victory for Kyrgios came on the same day that his lawyer asked an Australian court to dismiss assault charges against him on mental health grounds. Magistrate Glenn Theakston adjourned the case until Feb. 3.
The 27-year-old Kyrgios will appear in court in person on that date for the first time since he was charged by police by summons in July.
The common assault charge, which has a potential maximum sentence of two years in prison, relates to an incident in January 2021 that was reported to local police last December. The charge reportedly relates to an incident involving his former girlfriend.