The fifth-seeded Williams, who lost the 2003 final to her sister Serena, made the fourth round or better at all four Grand Slams a year ago. Her match was delayed briefly in the final set for the roof at Hisense Arena to be closed when rain began.
Errani continued a losing trend, dropping her 18th straight match against top 10 players.
It was the 20-year-old Wozniacki's first victory of the year, following a straight-sets loss to Slovakia's Dominika Cibulkova last week in the Sydney International and exhibition losses to No. 2 Vera Zvonareva and Kim Clijsters in Singapore and Hong Kong.
Wozniacki didn't face a break point in the first set and converted her only chance.
Dulko, the top-ranked doubles player, tested Wozniacki in the second set and pulled level at 4-4 with a service break, but the Danish player broke back immediately and served it out on her second match point after a double-fault on her first.
Wozniacki won six tournaments in 2010, including four of her last six, and she can retain the top spot by reaching the semifinals.
"My dream when I was a little girl was to reach No. 1 in the world, and that happened last year. But it's a new year, a new start," she said in a courtside interview. "I'm just looking forward to it."
Sharapova, the former top-ranked player who missed the 2009 tournament because of a shoulder injury and was ousted in the first round last year by fellow Russian Maria Kirilenko, struggled with her serve and was inconsistent in the second set. After losing the opening service game at love, Sharapova won seven straight games to take a 6-1, 1-0 lead.
The 33-year-old Tamarine, clearly not in peak physical condition, rallied and had a game point on serve for a 4-1 lead, but Sharapova broke back and regained momentum, winning the last five games.
Even when she was winning, though, Sharapova had trouble finding range with her serve. She had 10 double-faults and five aces.
Nerves had a role in that.
"I definitely felt that in the beginning. I knew I had an early exit last year and didn't want that to happen this year," said Sharapova, who has the goal in Australia of "staying aggressive and little by little trying to get my game better, maybe to where it was a few years ago and maybe better than that."
Justine Henin overcame a slow start to beat Indian qualifier Sania Mirza 5-7, 6-3, 6-1. The 2010 Australian Open runner-up had one set point in the first, but allowed Mirza to break her service, then win the next two games to take the set.
The seven-time Grand Slam winner said it was a difficult match, but "even if I wasn't at my best, I was fighting at every point."
The sixth-seeded Schiavone had a chance to serve for the match at 5-3 in the third but was broken. The Italian player recovered quickly, breaking Parra Santonja's service in the next game to take the match.
No. 20 Kaia Kanepi of Estonia and No. 29 Dominika Cibulkova of Slovakia advanced, along with No. 30 Andrea Petkovic, who beat American Jill Craybas 6-1, 6-2. Regina Kulikova of Russia beat 2008 semifinalist Daniela Hantuchova 7-6 (3), 3-6, 9-7, making the 28th-seeded Slovakian the only women's seeded player to lose on the opening day.
Information from The Associated Press was used in this report.