Tennis, unlike many other sports, did not introduce a blanket ban on players from Russia and its ally Belarus after the invasion, which Moscow calls a "special military operation".
Some sports have recently begun to readmit Russian and Belarusian athletes following recommendations issued by the International Olympic Committee (IOC) last month.
Wimbledon banned players from the two countries last year but said in March it would accept them as neutral athletes.
"I was really sad to miss Wimbledon last year - of course for a reason, but it still was painful," Kasatkina told reporters after beating Ukraine's Lesia Tsurenko at the Madrid Open.
"I'm happy that we will be able to come back this year and to be honest we are (in) the luckiest sport as we are able to compete still."
Kasatkina, who reached the Wimbledon quarter-finals in 2018, added: "95% of the athletes from Russia could not go outside and compete in the international events, and we really appreciate this opportunity and that we can be on the international stage".
Tsurenko, who last year said she did not want to play Russians or Belarusians on the Tour, did not shake hands with Kasatkina after the third-round match in Madrid.
"The saddest part is the war is still going on, so of course the players from Ukraine have got a lot of reasons to not shake our hands," Kasatkina said.
"I accept it, and it is how it is. It's a very sad situation."
Kasatkina next faces compatriot Veronika Kudermetova later on Monday.
