Kyrgios, who defeated world number two Briton Andy Murray during his four-match winning run this week, brushed aside the challenge from Dolgopolov 6-3 6-4 after the 36th-ranked Gavrilova had opened with a 6-4 7-6(6) win over Svitolina.
Australia's only previous Hopman Cup title came in 1999 while the last time they reached the final was 13 years ago.
The victories at the Perth event will boost world number 30 Kyrgios' confidence as he prepares for the first grand slam of the season beginning in Melbourne on Jan. 18.
(Reporting by Sudipto Ganguly in Mumbai; Editing by John O'Brien)
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