Federer is chasing a record eighth Wimbledon title and he was already well on top at the start of his 19th All England Club campaign when Alexandr Dolgopolov was forced to quit with an ankle injury.
The world number three raced into a 6-3, 3-0 in 43 minutes on Centre Court before the Ukrainian limped off.
Federer, who hit his 10,000th career ace in the eighth game of the first set, will face either Dusan Lajovic of Serbia or Greek qualifier Stefanos Tsitsipas for a place in the last 32.
"It feels great to be back on Centre Court," said Federer, who recorded his 85th match win at Wimbledon, passing the mark he shared with Jimmy Connors.
Having unexpectedly won the Australian Open in January before taking the clay-court season off to rest, Federer, making a record-equalling 70th Grand Slam appearance, is the favourite to take the Wimbledon crown for the first time since 2012.
"Everybody's got a chance to win Wimbledon and for me it's no different. I already achieved my dream to be back here healthy. Now we'll see how far I can go," Federer said.
Djokovic, a three-time Wimbledon champion, was leading 6-3, 2-0 when Slovakian world number 47 Martin Klizan retired after just 40 minutes on Centre Court due to a calf problem.
The 30-year-old Serb will face Adam Pavlasek of the Czech Republic for a place in the last 32.
It was the kind of trouble-free first round Djokovic would have been hoping for after the most turbulent period of his career, the 12-time major winner losing in the Australian Open second round and the French Open quarter-finals.
"He had issues walking on to court. I tried to focus on my game plan, I was serving well and when it mattered I made a break," said Djokovic, who is aiming to win a first Grand Slam title since last year's French Open.
"It was great to be back on Centre Court. It's the cradle of tennis history, but you never like to end a match this way."
With defending champion Serena Williams sidelined while she prepares to give birth to her first child, the race to win the women's title is the most wide open in a generation.
World number one Angelique Kerber, the 2016 Australian and US Open winner, is among the favourites after reaching last year's final, which she lost to Serena.
The German has failed to maintain that form this season and her poor run culminated in an embarrassing French Open first round loss to Ekaterina Makarova last month.
But the top seed showed glimpses of her best form as she saw off American qualifier Irina Falconi 6-4, 6-4 in 87 minutes on Centre Court.
"I'm happy to be back, but this year is completely different for me. I'm just happy I'm through the first round," Kerber said.