Mannarino broke to love for a 5-4 lead to quickly establish early superiority.
However, the 19-time grand slam champion rallied to triumph 4-6 6-1 6-3 and set up a last-four tie against either David Goffin or Jack Sock.
The Swiss veteran fired 12 aces and advanced to a semi-final against the victor from American Jack Sock and third seed David Goffin.
Federer never faced a break point in the 61-minute match and won every point when his first service landed safely.
He romped to the second set after losing the first and then got over the line in the third breaking in the final game to seal the victory.
Del Potro has been on a good run of form in recent weeks, reaching the Shanghai Masters semi-finals and winning the Stockholm title last weekend.
Sloppy from the start, Federer is chasing an eighth title in his hometown tournament.
He was a distant 47th in the race in late August and his efforts appeared to take their toll on Friday as he suffered a back niggle which required a visit from the trainer. "I've been playing a lot of matches".
"But I'm not putting my body at risk if I play tomorrow or next week (Paris Bercy)". I face a tough opponent tomorrow (Roberto Bautista Agut), so we will see what will happen.
The fourth seed, who won the Basel title in 2012 and 2013, came through in straight sets against Frenchman Benneteau 6-4 6-4.
Also Thursday, Marin Cilic, the second-seeded defending champion, beat 54th-ranked Borna Coric 6-3, 3-6, 6-3 to improve to 5-0 against his Croatian compatriot.