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Ostapenko, looking to add to the fifth career title she won in Dubai 12 months ago, edged an entertaining contest 7-5 1-6 7-5 in two hours and 19 minutes.
Fellow seeded player Anett Kontaveit was also victorious on the opening day of the tournament, but only after battling back to beat Zhang Shuai 4-6 7-6 (7-4) 6-4.
Marta Kostyuk saw off Sorana Cirstea 6-2 1-6 6-2, meanwhile, and will now face second seed Belinda Bencic in the next round.
Elise Mertens is also through to round two, where Veronika Kudermetova awaits, after dominating Martina Trevisan 6-0 6-2.
In the first main draw match at the Linz Open, Sofia Kenin let an early lead slip to lose 2-6 6-3 6-4 to Jule Niemeier.
