The World No.5 from China beat America's Bethanie Mattek-Sands in straight sets 6-4 6-3.
Last year he lost in straight sets just days after the death of his grandfather.
In the early matches, fifth-seeded Amelie Mauresmo eliminated Russian Tatiana Panova in straight sets 6-4 6-3.
Del Potro beat compatriot Leonardo Mayer in straight sets to reach the fourth round.
Serena Williams is already into the second round with a straight sets win over Varvara Lepchenko.
Sharapova, the reigning French Open champion, lost to Serena Williams in straight sets in the Spanish capital.
Second seed Nadal defeated Gael Monflis in straight sets, to book his place against compatriot Nicolas Almagro.
I've won my matches in straight sets but it's definitely a lot harder than the scoreline suggest.
Tomas Berdych set up a third-round match with Verdasco after beating Richard Gasquet of France in straight sets.
Federer will next face Swiss compatriot Stanislas Wawrinka after he disposed of American Andy Roddick in straight sets.
Tomas Berdych beat Jiri Novak in straight sets to advance while Belgian Kristof Vliegen upset compatriot Christophe Rochus.
Sharapova, beaten by Azarenka in the Australian Open final, also saw off Makarova in straight sets in Melbourne.
Kuznetsova, meanwhile, will now play fellow Russian Kirilenko after the 30th seed beat Yvonne Meusburger in straight sets.
Sweden's Robin Soderling progressed in straight sets as he overcame the serve of Janko Tipsarevic 6-3 7-6 (7-5).
Earlier Zheng's compatriot Li Na followed her into the quarterfinals after stunning fourth seed Caroline Wozniacki in straight sets.
The top seed also beat Willstrop in straight sets to win the Commonwealth Games title in Delhi last October.
But Davydenko had recovered from his opening match defeat to Djokovic by crushing Nadal in straight sets on Wednesday.
Over in Prague, the Czech Republic took a 2-1 lead against Serbia after winning the doubles in straight sets.
The 25-year-old Djokovic will play Ukrainian 14th seed Alexander Dolgopolov, who dispatched Stanislas Wawrinka of Switzerland in straight sets.
Golding took gold in the boys' tennis doubles after he and Czech partner Jiri Vesely triumphed in straight sets.
Kevin Anderson beat John Isner in straight sets to draw the South Africans level before the concluding mixed doubles.
The Serb will face Italy's Fabio Fognini in the last four after he defeated France's Richard Gasquet in straight sets.
American Sam Querrey booked his place in the last four following a straight sets 6-3 7-5 win over Xavier Malisse.
Zheng's compatriot Li Na followed her into the quarterfinals on Monday after stunning fourth-seeded Dane Caroline Wozniacki in straight sets.
Hawkeye's replay technology also played a part in Federer's victory because without it he would have won in straight sets.
Second seed Juan Martin del Potro cruised into the semifinals with a victory in straight sets against Finland's Jarkko Nieminen.
He will now play American youngster Jack Sock after the 20-year-old defeated Spain's Guillermo Garcia-Lopez in straight sets, 6-2 6-2 7-5.
Melzer will face last year's runner-up Milos Raonic on Sunday after the in-form Canadian defeated Germany's Benjamin Becker in straight sets.
However, Djokovic beat Federer in straight sets the last time they met at the Australian Open, in the semi-finals in 2008.
BBC: Novak Djokovic makes short work of Berdych in Melbourne
Earlier, unseeded Clijsters beat China's Li Na in straight sets to continue her incredible comeback to tennis after a two-year layoff.
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