Federer, who won in a stroll over Rafael Nadal just over a week ago in the Shanghai final, continued his top form in front of the local crowd at an event he has won seven times.
The 36-year-old Swiss, who has competed in a dozen Basel finals, made quick work of his 19-year-old opponent, who had taken him to five sets in this year's US Open first round.
Federer will next face France's Benoit Paire, a 6-3, 7-6 (7/4) winner over American Steve Johnson.
“I felt good throughout. The short match didn't matter too much because I have a day off tomorrow. Even if it had been three sets, it would have been OK,” Federer said.
“As long as I'm winning, it's all good. I'm just happy to win the first match. Frances was tough in the US Open.
“The next round with Paire will be tough, he's a dangerous player indoors. I'm happy I've found my rhythm early in Basel.”
Defending champion and second seed Marin Cilic advanced in just 30 minutes after German Florian Mayer quit when trailing 6-3.
Third seed David Goffin defeated German Peter Gojowczyk 6-2, 7-5 and fifth seed Jack Sock of the US rallied past Canadian Vasek Pospisil 3-6, 7-6 (9/7), 7-5.
Argentine Leonardo Mayer knocked out number eight seed Mischa Zverev 7-5, 7-5.
Against Tiafoe, whom he also defeated last March in Miami, Federer wasted no time in establishing domination of the young gun.
The Swiss quickly rolled 4-0 ahead, with Tiafoe finally earning a game after 21 minutes of play.
Federer then produced a love game and closed out the set in 29 minutes on his first set point.
Tiafoe's resistance stiffened slightly in the second set, with Federer achieving the break in the eighth game for 5-3 before wrapping up victory in 61 minutes.