Returning to Davis Cup tennis, Federer cruised past Ilija Bozoljac, winning 6-4, 7-5, 6-2 in less than two hours. Federer faced little challenge from Bozoljac, who served well but still was no match for his Swiss opponent.
Australian Open champion Wawrinka faced a tougher task against 102-ranked Dusan Lajovic in the second match, before winning 6-4, 4-6, 6-1, 7-6 (7).
Bozoljac began his match against Federer with two aces and had a set point while leading 5-2 in the second. But Federer held serve and broke to regain control over the set.
"Ilija was playing well, especially in the second set," Federer said. "It was a tough game, it could have been one set all."
Federer was making his first Davis Cup appearance since 2012.
Bozoljac said he was disappointed about his missed chance to win the second set.
"I lost focus and Federer stepped up the game," Bozoljac said.
Lajovic opened his match with a break, but Wawrinka soon broke back and went on to win the set easily. Lajovic then took advantage of several unforced errors by Wawrinka to take the second set.
However, Lajovic missed two set points in the tie break of the fourth before Wawrinka closed out the match.
"It was a tough game, he played well," Wawrinka said. "It's never easy when you come so late after an important tournament, but I managed to overcome that."
Switzerland is an overwhelming favorite against Serbia, last year's finalist and the 2010 Davis Cup champion, which is playing without Novak Djokovic and two other top players.
In Friday's other first-round matches:
The ninth-ranked Gasquet had little trouble beating teenager Nick Kyrgios 7-6 (3), 6-2, 6-2, and the 10th-ranked Tsonga followed up just as emphatically to dispatch Lleyton Hewitt 6-3, 6-2, 7-6 (2) and improve to 4-0 in head-to-heads against the former two-time Grand Slam champion.
"They were two very good matches," France captain Arnaud Clement said. "They did what they had to do and played the right way."
Berdych, an Australian Open semifinalist, dominated throughout as he cruised to a 6-3, 6-3, 6-0 victory for the defending champions in the first round matchup.
Never facing a break point, the seventh-ranked Berdych broke Sijsling early in each set and secured one break in the first set, two in the second and three in the third before converting his first match point with a powerful forehand winner.
"I tried to go on attack," Berdych said. "I'm glad that I got rid of the fatigue (from the Australian Open). I felt really good on the court and played accordingly."
The 46th-ranked Haase won the hard fought contest 3-6, 6-4, 6-7 (4), 6-2, 6-1 in 3 hours and 39 minutes on an indoor hard court at CEZ Arena.
Kohlschreiber had little difficulty in defeating Roberto Bautista Agut 6-2, 6-4, 6-2 in 1 hour, 43 minutes.
The Associated Press contributed to this report.