Wozniacki, who finished back-to-back years at No. 1 in the rankings in 2010 and `11, has never won a major and is determined to make up for a first-round exit at the Australian Open last year.
Puig fought back to win the second set and force a decider but the Rio champion eventually succumbed after 2 hours and 10 minutes.
Third-seeded Dominika Cibulkova, last year’s WTA Finals champion and a finalist at the Australian Open in 2014, opened in Sydney with a 6-2, 6-0 win over Laura Siegemund, advancing along with No. 6-seeded Johanna Konta and No. 9 Roberta Vinci.
Americans Christina McHale and CoCo Vandeweghe had wins over seeded players, with McHale beating No. 4 Kateryna Bondarenko 4-6, 7-5, 6-2 and Vandeweghe leading 6-2, 4-0 when No. 8 Elena Vesnina retired from their match.
Australian hope Sam Stosur, the 2011 U.S. Open champion, had another first-round defeat, losing 6-3, 6-1 to Anastasia Pavlyuchenkova in 63 minutes.
On the men’s side, fifth-seeded Philipp Kohlschreiber beat Fabio Fognini 6-4, 6-4, Matthew Barton beat Kyle Edmund 7-6 (3), 7-6 (5) and Daniel Evans had a 6-3, 4-6, 6-3 win over Thiago Monteiro.
At the Hobart International, Andrew Petkovic opened with a 6-4, 6-0 win over Nicole Gibbs and No. 11 Johanna Larsson beat Kristyna Pliskova 6-7 (5), 6-3, 6-1. Lucie Safarova also advanced, while former French Open champion Francesca Schiavone lost to Jana Jett and No. 7-seeded Kristina Mladenovic was beaten by Elise Mertens.
At Auckland, New Zealand, seventh-seeded Steve Johnson overcame an early struggle in a match of contrasting styles to beat Stephane Robert of France 6-4, 6-3 in the opening round of the ASB Classic.
Johnson struggled to bring his heavy serve to bear at the start of the match and quickly found himself a break down at 3-1 in the first set. He also took time to find his feet against Robert who troubled him with his softer groundstrokes, deft touches and his thread-the-needle passing shots.
”At the start of the year you don’t have much momentum and you try to figure out your game again and how everything works,” Johnson said. ”He came out and played a great game.
”I was in a lot of games on his serve in the first so I just sort of hung around and then fortunately got a break in the second and held onto my serve.”
The match pivoted on eighth game when Johnson was finally able to draw level on his sixth breakpoint chance. He then began to serve more confidently, taking the ninth game with his sixth ace of the match and breaking Robert again to end the set.
Sixth-seeded Feliciano Lopez rebounded to beat No. 1,035-ranked local wild-card entry Michael Venus 3-6, 6-4, 6-3.
Also advancing were Jiri Vesely, who won his only ATP title to date in Auckland in 2015, and Robin Haase.