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Eighth seed Millman was made to work exceedingly hard to avoid an early exit, eventually coming through a contest in which the two Australians faced a combined 31 break points by a 7-6 (7-3) 3-6 7-5 scoreline.
Ricardas Berankis won a 22-point tie-break in the first set to defeat Maximilian Marterer, while there were also wins for Millman's next opponent Guido Pella, Federico Coria and Ilya Ivashka.
At the Estoril Open, Kevin Anderson saved a match point at 6-5 down in the second set tie-break and turned around a 4-1 deficit in the third set to beat Frances Tiafoe 4-6 7-6 (8-6) 7-6 (7-4).
"It was obviously a very, very close match," Anderson said. "I think there were a lot of positives, a lot of really high-quality tennis, especially from midway through the second until the end of the match, I was really happy with the way I played.
"I think even if I had lost that match, obviously it's always tough losing, but I definitely felt I took a step in the right direction.
"Getting the win is a big added bonus, it gives me another opportunity to come out in a couple days' time."
Eighth seed Alejandro Davidovich Fokina beat Pablo Andujar in three sets, with Jeremy Chardy and qualifier Nuno Borges also victorious in Portugal.