Milos Raonic has won the last of his eight ATP Tour titles last year in Brisbane, Australia.
USA last won the title in 2007, while Australia hasn't won the international men's team event since 2003.
Top-seeded Kei Nishikori was ousted in the first round of the clay-court tournament by local favorite Thomaz Bellucci.
The 21-year-old Medvedev rallied from a double break down in the decider after breaking his French opponent when Paire served for the match in the second set.
Bellis is one of two teenagers in the quarterfinals with 19-year-old Ana Konjuh of Croatia beating eighth-seeded Elena Vesnina of Russia 3-6, 6-4, 7-6 (4).
Nicolas Mahut withstood 10 aces to beat German teen Alexander Zverev 7-6 (5), 7-6 (5) and make it into the round of 16 in Open 13.
Kei Nishikori crashed out of Rio de Janeiro Open after suffering a 6-4 6-3 first-round loss to Thomaz Bellucci.
Kei Nishikori enters Rio Open as top seed but tennis will take back seat in Rio during the Carnival week.
Caroline Wozniacki won her first-round match in Dubai Championships over Daria Kasatkina of Russia 6-2, 7-5.
Roger Federer, the world number nine, is due to figure with sixth-ranked Rafa Nadal at the Laver Cup.
Bjorn Bjorg will captain Europe while John McEnroe will do the same for the opponents.
This victory is expected to move Ryan Harrison to No. 43 in the world, matching his career-best ranking reached in July 2012.