As women's second seed Vera Zvonareva also progressed, Clijsters was made to work by the awkward Cornet, who fought hard on her 21st birthday but could not prevent the Belgian's 7-6 (7/3), 6-3 win.
US Open champion Clijsters has a chance to return to world number one, two years after her long break to have a baby, if she wins here and top seed Caroline Wozniacki falls before the semis.
"I know (Cornet) didn't get the result she wanted, but it's not a bad place to celebrate your birthday out here on the Rod Laver Arena," Clijsters said.
"I hope she enjoyed it.
Both Soderling and Murray, the world number four and five, made it into the second week without dropping a set as they steamed towards a possible quarter-final showdown.
Murray, last year's beaten finalist, dismissed Guillermo Garcia-Lopez of Spain 6-1, 6-1, 6-2, as he seeks Britain's first Grand Slam title in 75 years.
Murray, 23, will play Jurgen Melzer in the next round and may then face a date with in-form Swede Soderling, who cruised through 6-3, 6-1, 6-4 against Jan Hernych.
"I have played well in the first three matches and it's tough to concentrate in this heat. It was a lot harder than the scoreline suggests," Murray said.
Soderling, who is receiving texts and phone calls from Bjorn Borg, is now eyeing his fourth successive Grand Slam quarter-final. He will next play Ukraine's Alexandr Dolgopolov, who beat 2008 finalist Jo-Wilfried Tsonga.
Milos Raonic, the 20-year-old Canadian with the thunderous serve, shocked 10th seed Mikhail Youzhny to reach the fourth round on debut, and register his first win against a top-10 player.
Montenegro-born Raonic, whose 230 kilometres (143 miles) screamer against Michael Llodra was the tournament's fastest serve, beat Russia's Youzhny 6-4, 7-5, 4-6, 6-4. Spain's David Ferrer also beat Lithuanian Richard Berankis, 20.
Croatia's Marin Cilic outlasted fellow five-set specialist John Isner, who played tennis's longest match at Wimbledon last year, 4-6, 6-2, 6-7 (5/7), 7-6 (7/2), 9-7 in 4hr 33min.
Austria's Melzer, the 11th seed, went through when 2006 finalist Marcos Baghdatis pulled out of their match with a finger injury while trailing in the fifth set.
Meanwhile, Russia's Zvonareva, who could also finish this tournament as world number one, survived a Lucie Safarova fightback to stay on course for her third successive Grand Slam final.
Zvonareva looked set for an easy win until she tightened up and needed two match points in a tense tie-break before going through 6-3, 7-6 (11/9) thanks to a forehand error from the Czech.
"Today I had my chances and didn't use them, and then we had to go into this long second set," Zvonareva said. "That's something that I will definitely have to improve for the next matches."
China's Peng Shuai beat Japan's Ayumi Morita to join compatriot Li Na in the fourth round, stoking hopes they can match Li and Zheng Jie's run to last year's semis. Poland's Agnieszka Radwanska also beat Romanian Simona Halep.
But Czech player Petra Kvitova doused local hopes when she knocked out below-par French Open finalist Samantha Stosur, the fifth seed, 7-6 (7/5), 6-3.
Men's top seed Rafael Nadal was playing Australian teen Bernard Tomic as he continues his quest for the "Rafa Slam" which would unite all four Grand Slam titles for the first time since 1969.
Nadal won the French Open, Wimbledon and US Open last year after retiring hurt from his Australian Open quarter-final against Andy Murray with a knee injury.