Federer, chasing a sixth title on his home court, defeated Luxembourg`s Gilles Muller 6-2, 6-1 in the first round, pounding down a ninth ace for three match points and converting on the first to wrap up a flawless performance in 46 minutes.
The Swiss now stands 52-9 at the stadium where he got his start in the game as a ballboy two decades ago.
He has played the last eight Basel finals and appeared in the title match on 10 of 15 occasions.
Second seed Nadal shrugged off his impending date with appendix surgery by routing outclassed Pierre-Hugues Herbert of France 6-1, 6-1 to reach the quarter-finals.
Federer was glad to have saved his energy as he heads into what could be a blockbuster end of season comprising next week`s Paris Masters, the World Tour Finals, a Davis Cup final against France and a shot at taking back the number one ranking from Novak Djokovic.
“I didn`t feel like it would go fast, I thought it would be tough,” said Federer of his lightning victory.
“I didn`t have an early read on his serve, had a few tough service games, and put myself under some pressure.
“But I took my opportunity at 3-2. Once I was in the lead I got a sense of how to serve and how aggressive I needed to play.
“I took control from the baseline. Once I got the break in the second set, I knew it would be hard for him. It was a great first round for me.
“It was great not to have wasted any energy. I have a lot of tennis to play in the next few weeks. I felt the first match (in his title victory at Shanghai) for several days in my body.
“Now I can relax and and attack again tomorrow (against Uzbek Denis Istomin).”
Second seed Nadal, who last played the Basel event in 2004 and returned with a 0-2 record to the St Jakobshalle, needed just 56 minutes to beat French qualifier Herbert.
The Spaniard, who has ended a course of antibiotics for his appendix problem, took full advantage of 11 double-faults off the Frenchman`s serve in what was a frustrating afternoon for the 120th-ranked challenger.
“I feel better, but a little bit tired,” Nadal said. “I`ve had a few (summer) months without playing much tennis (wrist injury). And taking the antibiotics makes the body go down.
“But I`m happy to be in the tournament and happy to be in the quarter-finals.
“I never would have thought to win so quickly and get this far. It`s fantastic to do this well on my most difficult surface, without a lot of practise and with all the pressure that came in the first round.”
The world number three called his win “a positive result for me.”
“It`s important for me to do well here.”
Nadal next plays either 17-year-old Croatian Borna Coric or Andrey Golubev of Kazakhstan.
Coric, tipped for a bright future, upset Latvian sixth seed Ernests Gulbis 7-6 (7/2), 6-3 while Golubev advanced over Russian Teimuraz Gabashvili 6-3, 6-3.
Nadal`s victory was straightforward as he totally dominated Herbert, breaking the Frenchman five times and saving the only break points he faced in the final game of the win.
The nine-time French Open champion looked healthy and fit despite taking antibiotics for two weeks.
The 28-year-old will need to have surgery, with doctors unsure of how many more weeks he can go on avoiding the operation.
Belgian seventh seed David Goffin joined Nadal in the last eight as he beat Croatian Ivan Dodig 7-6 (7/0), 6-4.