Prior to the match, Sabalenka - who has repeatedly said on previous occasions "no-one supports war" - had said she understood if Kostyuk "hates" her.
Kostyuk refused to shake hands with Belarus' Victoria Azarenka at last year's US Open and also said seeing Russian flags being waved by supporters of Vladimir Putin at the Australian Open in January "hurt a lot".
Speaking at a post-match news conference, Sabalenka said passionately: "About the war situation, I said it many, many times: nobody in this world, Russian athletes or Belarusian athletes, supports the war. Nobody. How can we support the war?
"Of course we don't support war. If we could affect anyhow the war, if we could stop it, we would do it. But unfortunately, it's not in our hands."
However, Kostyuk criticised Sabalenka for not directly saying she opposed the war, and called on her to use her "big platform" after "rejecting her responsibility".
"She never says that she personally doesn't support this war," Kostyuk said.
"She should talk for herself I think, first of all. Then, talk about all the other athletes because I personally know athletes from tennis that support the war.
"To say nobody, I think is a little bit strong because I think you can only speak for yourself."
Sabalenka is one of the favourites for the title at Roland Garros and could become world number one if Iga Swiatek fails to reach the quarter-finals.
After a shaky start from Sabalenka, featuring two double faults, Kostyuk got the first break but it appeared to shock the 25-year-old into finding another level.
She broke back immediately to start a devastating run of six successive games to which Kostyuk had no answer.
Though she pulled a game back, Sabalenka continued to command play, saving two break points in what proved to be her final service game as she wrapped up the win on her first match point.
She will face fellow Belarusian Iryna Shymanovich, a qualifer, in the second round in Paris.
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