By Martyn Herman
Defending men's champion Djokovic, who has been training in the south of France rather than fine-tuning the grasscourt arts since his French Open title quest turned to dust, swept past Germany's Philipp Kohlschreiber 6-4 6-4 6-4.
American 20 times grand slam champion Williams, who like the Serb rarely schedules in any grasscourt tournaments before Wimbledon, also proved impressive after a slow start, beating Russian debutant Margarita Gasparyan 6-4 6-1 on Court One.
The reigning U.S. Open, Australian Open and French Open champion is now half way towards a rare "calendar" grand slam last achieved by Steffi Graf in 1988 and only six wins away from holding all four majors at the same time.
"Honestly, I don't think about it. But every time I come into press, you guys talk about it," Williams, like her sister Venus bidding for a sixth Wimbledon title, told a news conference when asked about a possible repetition of the "Serena Slam" she completed in 2003.
"Naturally it's definitely getting more on my mind than I want it to be. But it's also six matches away."
The only blot on her copybook was a code violation for an audible obscenity early in the first set.
World number one Djokovic's hopes of completing his own "career slam" were scuppered by Stanislas Wawrinka in the Roland Garros final earlier this month, but the Serb showed no lasting effects as he dispatched Kohlschreiber in clinical fashion.
Knowing exactly when to squeeze the throttle, Djokovic broke serve in the 10th game of each set to safely negotiate what had looked like a dangerous opening obstacle.
"That's where maybe I can have a mental edge over him. He missed a couple shots," Djokovic, bidding to emulate his coach Boris Becker and win a third Wimbledon title, said after easing past the world number 33 on Centre Court.
"Overall a great performance against a quality opponent."
Djokovic might have expected to be playing 2002 champion Lleyton Hewitt in round two but the gritty Australian's Wimbledon career was ended by Jarkko Nieminen.
Not that 34-year-old Hewitt went down timidly, rather scrapping tooth and nail as he has done throughout a career that took him to the summit of the sport.
Nieminen, one of 37 men aged 30 or over in the singles draw -- a Wimbledon record -- won 3-6 6-3 4-6 6-0 11-9.
"I was always going to leave it all out there, everything I had in the tank. I certainly did that," Hewitt, who will retire after his home grand slam in Melbourne next year, said.
It was a good day for seeds.
Fourth seed Wawrinka, looking to emulate the likes of Bjorn Borg and Rafa Nadal by claiming a French Open/Wimbledon double, cruised past Portugal's Joao Sousa 6-2 7-5 7-6(3).
Japan's Kei Nishikori, seeded five, scraped past Simone Bolelli of Italy 6-3 6-7(4) 6-2 3-6 6-3 while big-serving Canadian Milos Raonic, the number seven, ousted Spain's Daniel Gimeno-Traver in four sets.
Croatia's Marin Cilic, who joined the elite by winning the U.S. Open last year, beat Hiroki Moriya of Japan 6-3 6-2 7-6(4).
Russian Maria Sharapova screamed back to full volume after she coughed and spluttered out of the French Open a few weeks ago, the fourth seed looking revitalized in a 6-2 6-2 defeat of British wildcard Johanna Konta on Centre Court.
"The first match of Wimbledon is never the easiest and especially against an opponent who has had a good few weeks and is a crowd favorite," Sharapova said.
Serbia's Ana Ivanovic, the 2008 French Open champion, powered past China's Xu Yifan 6-1 6-1 while Victoria Azarenka of Belarus, another of the nine women's grand slam champions in the women's draw, beat Anett Kontaveit of Estonia 6-2 6-1.
Spain's Carla Suarez Navarro became the biggest name to fall, the ninth seed thrashed 6-2 6-0 by Latvian Jelena Ostapenko.
(Editing by Ed Osmond and Clare Lovell)