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Nick Kyrgios has battled his way to a three-set win against Russian Andrey Rublev at the Kremlin Cup while Daria Gavrilova advanced in the women's draw.

Australian star Nick Kyrgios has returned to winning ways after his limp performance at last week's Shanghai Masters once again raised questions about his effort in ATP matches.

He toughed it out for an hour and 29 minutes in a 6-3 4-6 6-4 first-round win at the Kremlin Cup in Moscow against local Andrey Rublev.

Kyrgios will play Bosnian Mirza Basic in the second round.

At the Stockholm Open, John Millman came back from a set down to defeat fellow Australian Alexei Popyrin 4-6 6-3 6-2 in the first-round but Alex de Minaur and Matt Ebden were eliminated.

Millman advanced to a showdown with Greek third seed Stefanos Tsitsipas, who had an opening-round bye.

Australia's top ranked player de Minaur shared the opening two sets with Philipp Kohlschreiber, both going to tiebreakers, before the German veteran rolled through the decider to record a 7-6 (7-3) 6-7 (2-7) 6-1 victory.

Ebden, a quarter-finalist in Shanghai last week, was no match for veteran left-hander Fernando Verdasco.

The Spanish eighth seed cruised to a 6-4 6-2 win.

In the women's draw at the Kremlin Cup, Daria Gavrilova came through Tuesday's late match against lucky loser Vitalia Diatchenko.

The world No.38 Australian was pushed in both sets by the Russian in her 7-6 (7-5) 7-5 victory.

Gavrilova will meet British No.1 Johanna Konta in the round of 16.