Playing four days after his five-set loss to Juan Martin del Potro in the U.S. Open final, Federer shook off two break points in the opening game and was never in trouble against the 64th-ranked Bolelli.
Federer, who struggled with his serve in the U.S. Open final, had 10 aces to go with 35 winners.
"With the jet lag and surfaces change and playing here in Italy away, I knew it was always going to be a tough match," said Federer, who improved his Davis Cup record to 36-11.
"We knew these were key matches, and that we were able to get both gives us a great opportunity on the weekend," Federer said.
Seppi and Bolelli wore black armbands to mourn six Italian soldiers killed in a bomb blast in Afghanistan on Thursday.
The winner of the best-of-five series will return to the World Group next year.