She narrowly missed making it four in a row, but claimed victory when Zheng punched a backhand into the net.
Zheng, a doubles expert, then produced a stunning run to the Wimbledon semifinals in 2008.
Last year, Zheng signed a lucrative endorsement deal with ANTA, a sportswear company in China with ambitions to break out overseas.
As play raged on, Celtic were awarded a penalty when Papac made a clumsy challenge on Zheng Zhi just inside the box.
Zheng took a 4-2 lead in the second set but Dulgheru then strung together a run of four straight games to keep her grip on the trophy for the second year.
In a recent simulation, Zheng Song of the University of Chicago, Fabrizio Zilibotti of the University of Zurich and two colleagues show that pensions need to be cut by over 35% to bring contributions and payouts into line over the long run.
ECONOMIST: China is beginning to face up to its pension problems
Referee Craig Thomson awarded Celtic a penalty after Sasa Papac fouled debutant Zheng Zhi, but the visitors also claimed for a spot kick when David Weir appeared to foul Shaun Maloney.
Last year's U.S. Open finalist Wozniacki defeated China's Zheng Jie, a semifinalist at the Australian Open in January, 6-4 4-6 6-1 in a match lasting more than two hours later on Wednesday.
CNN: Radwanska to face friend Wozniacki in Indian Wells semis
And when the markets are up, a fund's returns don't necessarily tell you anything about how it will perform in the future, says Zheng Sun, a finance professor at the University of California, Irvine.
The current problems have accumulated during the past several decades and there is no quick solution to the problems, Zheng Gongcheng, a professor at Renmin University of China, was quoted saying in China Daily.
But Sharapova regrouped in the second set and though Zheng battled back from a break down to draw level at 4-4, the Russian broke again to take the match into a decider, which she won comfortably.
She overcame eight double-faults, the erroneous ruling and a second-set deficit to beat Zheng Jie 6-1, 7-5 in the third round Saturday at the French Open.
Paddy McCourt was given 20 minutes to fashion an equaliser for Celtic, replacing Zheng Zhi as the visitors sought a greater attacking threat.
Zheng is into the semifinals of a Grand Slam for the second time, following her efforts at Wimbledon in 2008 and is relishing a contest against one of her heroes.
Dulgheru raced into a 5-2 lead in the first set before Zheng broke back to only to lose her own service and the opener.
Wozniacki, who had received a bye into the second round, was beaten by Zheng's aggressive tactics with the Chinese hitting twice as many winners as the Dane (34 to 17).
But, on balance, Ms Zheng calculates that the boost to exports from a 12-15% devaluation might deliver an extra 1.1% of growth, even if offset against increased import and debt-service costs.
The compass was also invented in China, and records exist of how a great fleet was sent under the command of Admiral Zheng He in the early 15th century to show the Chinese imperial flag in the ports of the Indian Ocean and the Persian Gulf, a fleet said to be the largest seen in the world up to that time.
Zheng learned the rudiments of tennis growing up as a child in China's lush and rustic heartland.
It was the first time the 19-year-old had beaten 18th seed Zheng in four meetings, and also gave her a first semifinal berth at Indian Wells.
CNN: Radwanska to face friend Wozniacki in Indian Wells semis
Henin will now face China's Zheng Jie who cruised into the last four with a straight sets - 6-1 6-3 - victory over Russia's Maria Kirilenko.
"She's a very tough player and fought until the last point, " Zheng was quoted as saying by the Women's Tennis Association website.
Uncertainty about future economic conditions has added at least a percentage point to the unemployment rate, according to Sylvain Leduc and Zheng Liu of the Federal Reserve Bank of San Francisco.
Though Zheng lost, it was by far the best performance ever by a Chinese player at a major.
The Dane, now ranked 10th and a winner of the tournament in 2011, progressed with a 6-0 6-1 drubbing of China's former Wimbledon semifinalist Zheng Jie.
Ninth-seeded Samantha Stosur, who has spoken of her anxiety over playing in front of her home fans, served twice for the match but double-faulted on match point in a 6-4, 1-6, 7-5 loss to China's Zheng Jie.
Zheng, the world number 35, edged an epic 65-minute first set in a thrilling tie-break.
World number 35 Zheng made nine unforced errors compared with 20 by her Russian opponent who had struggled with a leg injury throughout the contest.
Sharapova had not played since losing to China's Zheng Jie at Indian Wells and her lack of match practice after an elbow injury showed in a 6-4 6-3 defeat at Caja Magica.
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