Agassi came from a set down to defeat the defending Australian Open champion from Sweden, 6-7 (2-7), 6-4, 6-3. The 32-year-old American is 43 points behind Australia's Lleyton Hewitt in the ATP champions race.
Agassi could become No. 1 if he outperforms Hewitt in this event. Both also can secure points in next month's season-ending Masters Cup in Shanghai, China.
"I think it's an incredible achievement for anyone to finish No. 1 at any stage of their career. I mean, for me to be in this position is an incredible feeling," Agassi said. "So I can only imagine how it will be to actually do it."
Agassi, a winner of 15 Masters Series titles, will face 10th seed Carlos Moya of Spain in Friday's quarterfinals. Moya survived six match points before defeating defending champion Sebastien Grosjean of France, the seventh seed, 3-6, 7-6 (12-10), 6-1.
The only player to save more match points this season has been Marcelo Rios, who fought off seven in a first-round match in Sydney.
"The match was very intense, very physical, and in the end of the match I had a letdown physically," Grosjean said.
Hewitt, the top seed, kept pace Thursday by defeating No. 14 Yevgeny Kafelnikov of Russia, 6-2, 7-6 (7-3). Hewitt next faces eighth seed Roger Federer of Switzerland, who beat Germany's Tommy Haas, 6-2, 7-6 (7-2).
Third seed Marat Safin of Russia, who clinched one of the remaining spots in Shanghai on Wednesday, defeated Sjeng Shalken of the Netherlands, 7-6, 7-5.
Only eight players qualify for the Tennis Masters Cup. Sixth seed Tin Henman of Britain missed his chance to qualify when he lost to France's Nicolas Escude, 6-4, 6-2.
American Andy Roddick can clinch a trip to Shanghai by advancing to the final here. The 20-year-old moved a step closer Thursday with a 7-6 (7-5), 6-4 win over Romania's Andrei Pavel. Roddick's next opponent will be Paradorn Srichaphan of Thailand, a 6-1, 6-4 winner over Argentina's Guillermo Canas.
Notable deaths of 2014 [PHOTOS]
UPI Sports Results for September 19, 2014