Nadal defeated South African Rik de Voest 6-0, 6-2 while Wozniacki rolled over American wildcard Sloane Stephens 6-3, 6-2 at the Indian Wells WTA and ATP Masters 1000 event.
Nadal, who helped Spain win its Davis Cup first round tie against Belgium last week, now has two career wins over de Voest but it's been eight years between matches.
The Spaniard won their only previous meeting in 2003 at a Challenger event in France when he was just 16.
"I am better player now. The worst thing is I am eight years older," said the top seed Nadal, who had a bye in the first round.
He will next tackle Ryan Sweeting who beat Argentine 27th seed Juan Monaco 6-1, 0-6, 6-1.
Wozniacki also had little trouble, dispatching Stephens with a steady serve and precision forehand to win in just 68 minutes, wrapping up victory with a running forehand volley.
American qualifier Donald Young shocked world number five Murray 7-6 (7/4), 6-3 with the Scotsman looking rusty in his first career meeting against his 21-year-old opponent.
"I started the match well but then when it got close I didn't do anything to lift myself," said Murray.
"The crowd was for him and then he started to play better and I didn't play my way back into the match."
It marked the first victory over a top-10 opponent for Young.
"I am still thinking about it," he said. "I've been in this position a couple of times with a few players and had not won the match.
"I just told myself this time I was going to see it through and not let the nerves get the best of me."
Spain's sixth seed David Ferrer was also sent crashing, falling in the second-round to unseeded Ivo Karlovic 7-6 (7/3), 6-3.
The 32-year-old Croatian, who last week set a new world record with a 251kmh serve in his country's Davis Cup tie against Germany, hammered nine aces against Ferrer who had already won titles this year in Auckland and Acapulco.
Karlovic closed out the 96-minute match with an ace to post his first win over Ferrer in three career meetings.
Argentina's Juan Martin del Potro, the former US Open champion, continued his impressive comeback from a wrist injury by rallying to beat defending champion Ivan Ljubicic 5-7, 6-4, 6-2.
"It's not an easy draw," Ljubicic said of meeting del Potro in his opening match.
"He's a great player, and it was a great match. We had some great points but I came up short in the end."
In the late match that ended after midnight local time, fourth seed Robin Soderling extended his record this year to 19-1 by beating Germany's Michael Berrer 6-3, 7-6 (7/4).
In the women's draw, seventh seeded Li Na of China lost to countrywoman Peng Shuai 4-6, 6-3, 6-3 while former US Open champion Svetlana Kuznetsova was beaten by American wildcard Christina McHale 7-6 (7/4), 7-6 (9/7).
Former world number one Dinara Safina of Russia ousted two-time Indian Wells champion Daniela Hantuchova 7-6 (7/2), 6-4.
Roger Federer, seeded number two, plays his first match Sunday against Russia's Igor Andreev.
Australian Open champion, and last year's runner-up, Novak Djokovic, Canadian wildcard Milos Raonic and women's second seed Kim Clijsters are all in action on Sunday.