Federer defeated Paire for the fifth time without a loss, mowing down the challenger in just 57 minutes in front of a sympathetic audience at the St Jakobshalle.
The 36-year-old Federer, winner of his last event in Shanghai this month, is seeking an ATP-leading seventh trophy this season as he aims for the weekend final in his hometown.
"I'm happy with how it went, I'm always ready for a tougher one," Federer said. "I got a good start, got rolling.
"I was able to put more presure on him than him on me. I've had a good two first rounds. I thought the draw looked tough with (Francis) Tiafoe and Paire in my section. I'm a little surprised, to be honest.
"It's not easy to win 1 and 3 indoors, this was a great start."
Federer ripped through the opening set and piled on more pressure in the second after Paire took a massage treatment on his back after losing the opener.
World number two Federer remained in command as he broke Paire to love in the seventh game, with the French player sending down his second double fault to drop the game.
Federer made it a comfortable 5-3, with Paire sending a return well wide.
The 19-time Grand Slam champion Swiss earned a match point in the ensuing game, which his opponent saved with a service winner.
But Federer didn't relent, his smash setting up a second match-winner followed by an untouchable forehand down the line to end the evening.
Federer will next face French seventh seed Adrian Mannarino who beat promising Canadian Denis Shapovalov 4-6, 6-1, 6-2.
Defending champion Marin Cilic beat Croatian compatriot Borna Coric 6-3, 3-6, 6-3 in a two-hour struggle to move to 5-0 against his younger countryman.
Fourth seed Juan Martin del Potro edged closer to a spot at next month's ATP Finals as he dispatched French qualifier Julien Benneteau 6-4, 6-4.
Del Potro won the Basel title in 2012 and 2013, beating Federer in both of those finals.