World number one and six-time champion Williams, bidding for a record-equalling 22nd Grand Slam title, battled to a 6-2, 6-4 win over Swiss qualifier Amra Sadikovic, ranked at 148 in the world.
She will face fellow American Christina McHale for a place in the last 32.
“I would be lying if I said I feel fresh, but I don’t feel fatigue. I feel real hungry, super motivated, extremely ready to do the best,” said the American star, whose mother Oracene Price was watching from the front row of the Centre Court Royal Box.
“I never underestimate any opponent. I always expect the best of myself,” added Williams, who won 12 straight points at the start of the match to open a 3-0 lead.
However, not everyone gave her the royal seal of approval.
“It was one of the worst serving efforts I’ve seen from Serena. This is what is giving other players hope,” said US legend John McEnroe, commentating for the BBC.
Top seed Williams won her 21st major at the All England Club 12 months ago by beating Garbine Muguruza.
But her hopes of going level with Steffi Graf’s Open era record of 22 has stalled since that day.
She went down to a US Open semi-final loss to Italy’s Roberta Vinci followed by defeat in the Australian Open final to Angelique Kerber and then to Muguruza in the French Open final.
– Tricky and breezy –
Murray, the 2013 champion, made a flying start to his bid for a second Wimbledon title as the world number two crushed compatriot Liam Broady 6-2, 6-3, 6-4.
The 29-year-old brushed aside the world number 235 and will next play Taiwan’s Lu Yen-Hsun.
“The first match is always tricky and it was pretty breezy out there at the beginning,” said second seed Murray.
“We’ve practised together in the past. It’s not easy playing someone you know, so I did ok.”
Minutes after Murray had completed victory, heavy rain brought a halt to action on all courts except Centre Court where the roof was closed. Play was eventually suspended for the day with only 37 of the scheduled 65 matches completed.
Swiss fourth seed Stan Wawrinka defeated American teenager Taylor Fritz 7-6 (7/4), 6-1, 6-7 (2/7), 6-4 to set-up a second round blockbuster against Juan Martin del Potro.
Injury-plagued del Potro, a former US Open champion, is playing in his first Grand Slam since the 2014 Australian Open after undergoing three wrist surgeries.
Del Potro made the second round by beating France’s Stephane Robert 6-1, 7-5, 6-0.
The Argentina star is ranked at 165 but made the semi-finals at Wimbledon in 2013 where he lost a five-setter to Novak Djokovic. That was his last appearance at the tournament.
“He has always been a great champion, a great player to watch also,” said two-time quarter-finalist Wawrinka of the towering del Potro.
“It’s going to be a tough match. He’s playing well. So it’s going to be interesting to see what’s going to happen.”
Nick Kyrgios, the Australian 15th seed, beat Czech wildcard Radek Stepanek, at 37 the oldest man in the draw, 6-4, 6-3, 6-7 (9/11), 6-1.
He next faces German wildcard Dustin Brown who got past Dusan Lajovic of Serbia 4-6, 6-3, 3-6, 6-3, 6-4.
Both Kyrgios and Brown have been headline makers in the past at Wimbledon after beating Rafael Nadal in 2014 and 2015 respectively.
Kyrgios had a mid-match fallout with chair umpire Mohamed Lahyani and even told his own entourage to get out of the court.
The match took place on the same Court Two where last year the 21-year-old was accused of ‘tanking’ against Richard Gasquet.
“The crowd likes the way I play here, I entertain. I’m comfortable here and they know the tennis I bring every day,” said Kyrgios, who was warned for bad language by Lahyani.
Meanwhile, Svetlana Kuznetsova defeated Carolia Wozniacki 7-5 6-4.