Federer's 6-4, 6-4, 6-2 win in less than two hours restored order for the 15,000 Swiss fans who saw Australian Open champion Stanislas Wawrinka surprisingly lose the opening singles match.
Wawrinka, ranked No. 3 after his first Grand Slam title, was beaten 7-6 (5), 6-2, 3-6, 7-6 (5) by 64th-ranked Andrey Golubev.
Federer and Wawrinka, doubles champions at the 2008 Beijing Olympics, are scheduled to team up Saturday against Kazakh doubles pair Aleksandr Nedovyesov and Evgeny Korolev.
"The weekend is not finished," Federer said in a courtside interview after beating the 56th-ranked Kukushkin. "I served well and played a solid match."
The winner will face Britain or Italy in the semifinals in September.
Switzerland is aiming to reach its first semifinal since 2003 in pursuit of a first Davis Cup title.
Wawrinka received a huge ovation stepping on court at the Palexpo indoor arena for his first match in Switzerland since becoming a Grand Slam champion in January.
However, there was a stunned near-silence when Golubev converted his sixth match point with an overhead winner.
The third-ranked Wawrinka acknowledged starting "a little bit tight, a little bit nervous."
"When I'm in that position I know that I don't move so well," he said, after making 70 unforced errors and converting just one of his 10 break points.
The Swiss No. 1 also smashed his racket on the hard-court surface in frustration during the second set.
Golubev, who lost to Wawrinka in the first round in Australia, overcame his early nerves to win a first-set tiebreak.
"I tried to be always aggressive, didn't give him too much time," he said.
Wawrinka, who played in a 5-0 loss against Kazakhstan in 2010, said the heavily favored Swiss would not be complacent.
"We didn't come here thinking it was going to be an easy tie," he said.