Japan's Osaka into US Open quarter-finals
Osaka, the fourth seed, powered past her 14th seeded counterpart 6-3, 6-4 in the late match in an empty Arthur Ashe Stadium at Flushing Meadows.
The 2018 US Open champion said keeping her emotions in check was key to dominating the 1hr 12 min match.
"For me it's very important because it's quite easy to get down on yourself, especially when you feel like you could have converted so many times," she explained.
"But I just told myself that she's a great player she's supposed to hold her serve so yeah just keep pushing."
Frustration had got the better of Osaka in the previous round when she threw her racquet during a tie-break.
Osaka said she hadn't watched live the incident with Novak Djokovic earlier in the day when he was disqualified for accidentally hitting a lines judge with his ball after losing a game.
She was sleeping ahead of her night match, but after becoming aware of it she said it served as a reminder to remain in control while on the court.
"That's definitely like a warning to never do that," Osaka said.
Kontaveit fought hard to hang in the contest, saving five match points before Osaka booked her place in the last eight.
Japan's Osaka will play 27-year-old Shelby Rogers for a place in the semi-finals after the American upset sixth seed Petra Kvitova in three sets Sunday.
Osaka, of Haitian and Japanese heritage, walked onto the court wearing a facemask emblazoned with the name of Trayvon Martin, an unarmed black youth shot and killed by a white neighborhood watch member in Florida in 2012.
Osaka, the 2019 Australian Open champion, is wearing different masks honoring victims of racial injustice and police brutality throughout the tournament.
She has already donned face coverings bearing the names of Breonna Taylor, Elijah McClain and Ahmaud Arbery.
Results from US Open Tennis Championships in New York on Sunday (x-denotes seed):
Men singles 4th rd
Pablo Carreno-Busta (ESP x20) bt Novak Djokovic (SRB x1) 6-5 default
Denis Shapovalov (CAN x12) bt David Goffin (BEL x7) 6-7 (0/7), 6-3, 6-4, 6-3
Borna Coric (CRO x27) bt Jordan Thompson (AUS) 7-5, 6-1, 6-3
Alexander Zverev (GER x5) bt Alejandro Davidovich (ESP) 6-2, 6-2, 6-1
Women Singles 4th rd
Jennifer Brady (USA x28) bt Angelique Kerber (GER x17) 6-1, 6-4
Yulia Putintseva (KAZ x23) bt Petra Martic (CRO x8) 6-3, 2-6, 6-4
Naomi Osaka (JPN x4) bt Anett Kontaveit (EST x14) 6-3, 6-4
Shelby Rogers (USA) bt Petra Kvitova (CZE x6) 7-6 (7/5), 3-6, 7-6 (8/6)