Svitolina had said she would not shake hands with Russian or Belarusian players at Roland Garros because of Russia's invasion of her country. Belarus has been used as a staging area for what Moscow calls a "special military operation".
Kasatkina has been one of few Russian players to openly speak out against the war, calling it "a full nightmare". On Sunday, she was supported by Russian popular singer Zemfira, who left the country because of her opposition to the invasion.
Zemfira was placed on a Russian Justice Ministry list of foreign agents in February on the grounds that she supported Ukraine and criticised the invasion.
After her 6-4 7-6(5) defeat, Kasatkina gave Svilotina a thumb up and Svitolina was also seen putting her thumb up. It was unclear whether the gesture was addressed to Kasatkina.
Svitolina, however, said: "Really thankful for her position that she took. Yeah, she's really brave person to say it publicly, that not so many players did. She's a brave one."
Kasatkina was however booed by the crowd.
"Leaving Paris with a very bitter feeling. All these days, after every match I’ve played in Paris I always appreciated and thanked the crowd for their support and being there for the players," the 26-year-old wrote on Twitter.
"But yesterday I was booed for just being respectful on my opponent's position not to shake hands. Me and Elina showed respect to each other after a tough match but leaving the court like that was the worst part of yesterday.
"Be better, love each other. Don’t spread hate."
Geopolitics have been at the centre of this year's French Open, with Aryna Sabalenka of Belarus skipping her post-match press conference twice, saying she felt unsafe after being grilled about the war.
Last week, two-time French Open champion Novak Djokovic started a controversy after writing on a camera lens "Kosovo is the heart of Serbia" amid unrest in the region.
