World number four Federer, the 17-time Grand Slam title winner, swept past 56th-ranked Mikhail Kukushkin, 6-4, 6-4, 6-2 to leave the tie level at 1-1 after the first day.
Earlier, Wawrinka, the world number three, was stunned by unfancied Andrey Golubev, the world's 64th ranked player, in a 7-6 (7/5), 6 2, 3-6, 7-6 (7/5) shock.
"It was a great match I think for me," an understated Golubev said after one of his biggest career wins.
"I played well, it was not easy but I'm happy that I won. The key was to try and be aggressive and not to give him too much time. I think I did good."
On Saturday, Wawrinka and Federer were to take on Aleksandr Nedovyesov and Evgeny Korolev in the doubles.
The eventual winners of the tie will face either Italy or Great Britain in September's semi-finals.
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