Federer, 31, swept to a 6-3, 6-2, 6-0 win on Centre Court to celebrate the 10th anniversary of his first triumph at the All England Club. The seven-time champion Swiss fired 32 winners and just six unforced errors as he clinched a sixth career victory over the 47th-ranked Romanian.
"It's the most special thing to be back on Centre Court," said Federer, playing in his 55th straight Grand Slam and seeking an 18th major. "I still enjoy it out there. It was a pleasure to play and I'm very happy to get the first round out of the way.
"I am not sure how much better I could have played. It was good to get it done quickly as it was cold and it released some of the pressure." Federer came into Wimbledon on the back of his first title of 2013 in Halle, the grasscourt win taking him level with John McEnroe on 77 career titles.
He next plays Sergiy Stakhovsky of the Ukraine for a place in the last 32. Maria Sharapova put her bitter public feud with Serena Williams to one side to reach the second round.
The 2004 Wimbledon champion was given a stern test by France's Kristina Mladenovic but the Russian third seed eventually triumphed 7-6 (7/5), 6-3 on Centre Court. "I'm really pleased with this performance," Sharapova said.
"My opponent played extremely well so I'm really happy to be in the second round." Second seed Azarenka, a semi-finalist in 2011 and 2012, moved into second round, beating Portugal's world number 106 Maria Joao Koehler 6-1, 6-2.
The 23-year-old Belarusian, the reigning Australian Open champion, broke down in tears after suffering a worrying fall in the second game of the second set on Court One as her right leg gave way in her service action.
Azarenka collapsed in crippling pain before her match was delayed by 10 minutes for her to have her right knee heavily strapped.
"It was such a shock. I felt my whole body collapse," said Azarenka, who eventually regained her composure to book a second round match-up with experienced Italian, Flavia Pennetta.
Italian fifth seed Sara Errani became the first major casualty of this year's Wimbledon when she slumped to a 6-3, 6-2 defeat to Puerto Rican teenager Monica Puig.
Errani, a semi-finalist at the French Open earlier this month and runner-up in Paris to Maria Sharapova in 2012, had gone into her first round clash carrying a groin injury which forced her out of Eastbourne last week.
Puig, 19, the world number 65, who is playing a grasscourt tournament for the first time as a professional, goes on to face Silvia Soler Espinosa of Spain.
Puig, who became the first Puerto Rican woman to reach the third round of a major at Roland Garros, took victory on a seventh match point. Her latest success was celebrated by Puerto Rican popstar Ricky Martin, who tweeted his congratulations.
Asked if she is now a national hero, Puig said: "Yeah, basically. The big sports over there are boxing, baseball and basketball. "Now that I've been doing well, tennis is starting to become bigger as well. Everybody was watching the match. Everybody woke up really early."
Later Monday, double champion Rafael Nadal, fresh from a record eighth French Open title and a potential quarter-final opponent of Federer, looks to erase the memory of last year's horror show.
Twelve months ago, he was knocked out in the second round by Czech outsider Lukas Rosol, a defeat which sent him off the tour for seven months to rest his troublesome knees.
But the 27-year-old has banished the doubters in emphatic fashion this year, winning seven titles in nine finals. Nadal begins his campaign against Belgium's Steve Darcis on Court One, but Rosol is already out, losing 6-3, 4-6, 7-6 (7/5), 6-7 (4/7), 6-4 against Germany's Julian Reister.
Last year's runner-up Andy Murray faces Germany's Benjamin Becker on Centre Court as he looks to end Britain's 77-year wait for a homegrown male winner at Wimbledon.