The Argentine world number 10, who denied a report by French newspaper l'Equipe that he tested positive for a banned substance during his run to this year's French Open final, appeared distracted against the 81st-ranked Baghdatis.
U.S. Open semi-finalist Robby Ginepri cruised into the quarter-finals with a solid 6-4, 6-4 victory over Luxembourg's Gilles Muller. Czech Radek Stepanek and American Taylor Dent reached the third round of the $860,000 tournament.
Puerta never looked comfortable despite saving three match points to take the second set tiebreak 13-11. Baghdatis made no mistake on his fourth match point at 6-5 in the third.
"I don't think the reports affected my game," shrugged Puerta. "The reports surprised me completely and I have taken legal steps. Today I give my opponent credit."
Puerta's exit was another blow to organisers after Australian Open champion Marat Safin, Briton Tim Henman and Argentines Gaston Gaudio and David Nalbandian pulled out through injury.
Third seed Ginepri, who lost to British teenager Andy Murray in the quarter-finals in Bangkok last week, swept aside Muller in just 70 minutes.
The American will play Bjorn Phau in the last eight after the German thrashed 10th seed Kenneth Carlsen of Denmark 6-1, 6-2.
Second seed Stepanek beat fellow Czech Jiri Vanek 7-5, 6-4 in a second round match, and Dent, seeded fourth, overpowered Russia's Dmitry Tursunov 6-3, 7-6.
Finland's Jarkko Nieminen, the seventh seed, pulverised former Australian Open runner-up Rainer Schuettler of Germany 6-1, 6-1.
Baghdatis will next play Croatia's fifth-seeded Mario Ancic who beat Frenchman Cyril Saulnier 6-2, 6-1.
In the women's draw, top seed Vera Zvonareva eased past Israel's Shahar Peer 6-3, 7-5.
The Russian will play eighth seed Sania Mirza in the quarter-finals after the Indian crushed Japan's Aiko Nakamura 6-1, 6-4.
Second seed Nicole Vaidisova of the Czech Republic easily beat Japan's Rika Fujiwara 6-4, 6-2, and number three Tatiana Golovin thrashed another Japanese player, Saori Obata, 6-1, 6-4.
Fourth-seeded Russian Maria Kirilenko, who won her first WTA title in Beijing last month, eased past Slovakian Henrieta Nagyova 6-4, 6-2 to reach the last eight.