The 36-year-old destroyed Slovakia's Lukas Lacko 6-4 6-4 6-1 in 90 minutes on Centre Court. "I felt good out there, less nerves than in the first round", Federer said.
Fabbiano won the second set with a break in game six and he also led 3-1 in the third before Wawrinka bounced back, earning his first break of the match at 2-3 and staying in touch with his rival before the weather sent the encounter into Thursday's schedule.
Federer didn't drop a point on his serve in the entire second set and kept that streak going until he was 30-0 up at 4-1 in the third, when Lacko sent a backhand victor down the line.
Federer was relentless as he lost just nine of 61 total points behind his intimidating service. "Some players can play best tennis".
Williams was almost as good, losing just five of 32 points on her first serve in a 6-1, 6-4 win over Viktoriya Tomova.
But she didn't look her age in making it 89 career match wins at Wimbledon by demolishing Dulgheru in the final two sets.
Outside of the Williams sisters, who have 12 Wimbledon titles between them, Keys is viewed as the next likely USA champion at the All England Club.
In the first round, she started by conceding a tiebreaker against 58th-ranked Johanna Larsson before taking 12 of the last 15 games.
French 17th seed Lucas Pouille fell short in five sets after fighting from two sets down to be beaten 6-4, 6-2, 6-7 (8-10), 3-6, 6-2 by Austria's Dennis Novak. "You go from it being on clay where it usually is bouncing higher, and then all of a sudden the ball doesn't come up at all and you have to be a lot lower".
After losing just one set on her way to the Wimbledon title a year ago, Garbine Muguruza picked up exactly where she left off, beating British wild card Naomi Broady 6-2, 7-5 on Tuesday.
In the men's draw, Canadian Milos Raonic overcame a hard-fought, triple-tie-break battle to defeat Australian John Millman 7-6 (7-4), 7-6 (7-4), 7-6 (7-4).
Karolina Pliskova advanced to the third round for the first time with a 6-3, 6-3 victory over Victoria Azarenka.