The 13-time Grand Slam champion overpowered Arn 6-1, 6-1 in 59 minutes on Saturday for her 17th consecutive win at the Australian Open.
The mismatch was so great that the crowd was muted, rousing themselves only when Arn won her two games and when Williams completed her third-round victory.
Arn double-faulted twice to end the match. As the players shook hands at the net, Williams looked briefly taken aback and smiled.
“I told her it was an honor to play against you,” the 32-year-old Arn said. “And she told me, ‘Oh, you are so sweet.’ I’m a big fan of hers. She’s the real No. 1.”
Williams, who racked up her 501st career match win, is hoping to become the second woman over age 30 to win the Australian title in the Open era.
“It makes me feel really good,” she said of her Hungarian opponent’s remarks. “I’m really proud of the work that I’ve been doing for so many years, all the hard work.”
Vania King’s loss to Ana Ivanovic left Williams as the only American player left in either singles draw.
Former Georgia standout John Isner lost Friday, the last American man to exit.
Coming off an injury-ravaged 18 months, Williams is seeded 12th. She hasn’t held the top ranking since 2010, the year she won the last of her Grand Slam titles.
Because of her ranking, Williams can’t take the No. 1 spot with a win at Melbourne. However, No. 2 Petra Kvitova, No. 3 Victoria Azarenka and No. 4 Maria Sharapova could walk away with the top ranking if they win the tournament.
Next up for Williams is unseeded Russian Ekaterina Makarova.