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.
"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.