Clijsters sailed past Israel's Shahar Peer 6-2, 6-1 to join China's Li Na in the women's semi-finals while Baghdatis booked his place in the men's quarter-finals with a 6-4, 6-7, 6-3 victory over Czech Jan Hernych.
Defending champion James Blake also advanced to the last eight with a comfortable 6-2, 6-4 win over Australian wildcard Nathan Healey while Russian lucky loser Evgeny Korolev kept his unlikely run going by beating Robby Ginepri 0-6, 6-4, 6-4.
Clijsters, who was runner-up to fellow Belgian Justine Henin-Hardenne at the 2004 Australian Open, beat Peer so easily that she was able to use the match as practice.
"There's still a lot of things I feel like I can improve on, but I didn't really get tested today," she told a news conference.
"When
"I served the best I have on the whole trip so that was good...things are gradually getting better and better."
Baghdatis, whose fairytale run to last year's Australian Open brought the tournament to life, was given a tough workout from Hernych.
The charismatic Cypriot surprised his followers when he arrived on Sydney's Olympic Park centre court without his trademark beard and briefly gave them cause to worry before raising his game in the final set.
"It wasn't easy. It was hot out there and it's the first time I've played in the heat for a while so it was good to have a match like that before the Open."
Baghdatis plays Carlos Moya in Thursday's quarter-finals after the former world number one saved a match point in his 4-6 7-6 6-3 win over fellow Spaniard Fernando Verdasco.