The unseeded Indian, playing with Protected Ranking, edged past the Canadian 6-7 (4) 7-6(7) 6-3 in the opening round of the Euro 127,000 grass court event.
Yuki was down a match point at 6-7 in the second set tie-breaker before he forced a third set and finally prevailed in the contest after two hours and 35 minutes.
Yuki had as many as seven break chances in the opening set against Polansky but could convert none.
"It was another battle. I had chances to break him in the first set but could not, then I was down in the second set too and saved a match point, so it was up and down. I am happy to be able to pull it off in the end. It was important as I lost a close match to him in the French Open," Yuki told PTI.
In the doubles event, Jeevan Nedunchezhiyan lost his opening round with Sergio Galdos. They lost 6-7 (5) 6-7(5) to Andrei Vasilevski and Hans Podlipnik-Castillo.
Paes and Shamasdin are up against the Australian pair of Matt Reid and John-Patrick Smith while Raja and Sharan will open their campaign against Victor Estrella Burgos and Darian King.
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