Williams was seeded No. 25 by the tournament, despite being ranked No. 181 in the WTA world rankings.
"I have such high expectations of myself,'' Williams said after picking up a victory in her first match at the All England Club in two years. "I don't go out there expecting to 'do well' or 'see what happens.' That's just not me.''
Williams' seeding has been a hot topic in the tennis community recently. The 23-time Grand Slam champion returned to competitive tennis earlier this year following the birth of her first child in September. Although WTA rules allow women who miss time because of a pregnancy to enter events based on their pre-absence ranking, there is no guarantee of a seeding. Williams was unseeded at the French Open earlier this year.
"Not only do I expect to win,'' she said, "I expect to win emphatically.''
Rus is ranked No. 105. Williams improved to 56-2 in her major career against players ranked outside the WTA's top 100.
Having struggled to establish a rhythm against her Swedish opponent in the opening set, the ninth-seeded Williams won 10 games in a row from 1-2 in the second to take control.
The 38-year-old American, the oldest woman in the draw, reached the 2017 final and lost to Garbine Muguruza.
Wozniacki took 59 minutes to complete the victory on Centre Court, finishing it with a service winner.
The second-seeded Dane has never been past the round of 16 at the All England Club but showed good form on grass by winning her second title on the surface at Eastbourne last week.
Sixteenth-seeded American Coco Vandeweghe took a fall in the opening set of her first-round match at Wimbledon and then stumbled out of the tournament. Vandeweghe, a two-time Grand Slam semifinalist and two-time quarterfinalist at the All England Club, lost 6-7 (3), 6-3, 8-6 to 42nd-ranked Katerina Siniakova of the Czech Republic.
In the tiebreaker, Vandeweghe was running forward to chase a short ball when her feet slipped out from under her. She slid feet-first into the net and stayed down for a bit before being checked on, but she continued to play.
"The tougher balance, for me, is to be able to spend time away from my son and be OK with taking, sometimes, time for myself, which is a struggle sometimes because I really want to spend every second with him,'' said Azarenka, who faces No. 7 seed Karolina Pliskova next. Pliskova ousted Harriet Dart earlier Monday.
The 19-year-old Swan defeated her 36th-ranked opponent in just 52 minutes to earn the first Grand Slam victory of her career.
The 204th-ranked Swan is one of five British women ages 19-23 who were given wild cards to Wimbledon.
The Associated Press contributed to this report.