Kunitsyn claimed a 6-7 (6-8), 7-6 (7-4), 6-3, 6-4 triumph over Somdev Devvarman that took more the 3 1/2 hours. Youzhny followed with a 6-4, 6-2, 6-3 romp over Rohan Bopanna.
Kunitsyn overcame 65 unforced errors and four service breaks in winning.
Youzhny committed just 13 unforced errors and didn't serve in his match. He broke Bopanna's serve five times.
Russia can lock up a Davis Cup quarterfinals slot Saturday in the doubles match. Kunitsyn is to team with Teimuraz Gasbashvili versus the Indians Mahesh Bhupathi and Leander Paes.
The schedule calls for reverse singles Sunday with Kunitsyn taking on Bopanna and Youzhny playing Devvarman.
India owns a 2-1 Davis Cup series edge over Russia but the teams haven't met since 1986.
The winner of the India-Russia series will be matched against either Argentina or Sweden in the next round.