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MIAMI—Survivng a furious fightback by one of the greatest comeback players of this or any other era, Caroline Wozniacki took a big step toward regaining her confidence and place near the very top of the WTA pecking order with a convincing 6-4, 6-4 quarterfinal win over Serena Williams.

Wozniacki, seeded fourth here at the Sony Ericsson Open, showed more baseline aggression and determination tonight than in her most recent clash with Williams—a 6-2, 6-4 loss at last year's U.S. Open.

That one was a desultory defeat that did little to sway those who felt that Wozniacki was the WTA No. 1 in name only. Perhaps her performance tonight will banish what demons lingered in her mind after that loss, and convince critics that she’s made from Grand Slam champion material. This one was a reputation restorer.

After winning a tight first set tonight, Wozniacki bolted to a 5-1 lead, only to see Serena slug and holler and will her way back into contention. Before it was over, Wozniacki was a mere two points from allowing Serena to deadlock the second set at 5-all. But she held firm and prevailed thanks to a pair of backhand errors (the first on a solid Wozniacki serve, the second after a brief rally) by Serena.

Williams struggled to control her groundstrokes for the entire match. She made 36 unforced errors—a large number of them forehand drives right into the net—compared to just 13 by Wozniacki. But she also hit 34 winners, as compared to 12 by Wozniacki. And that serve that produced 20 aces for Serena in her last match? Tonight she hit just one more ace—four—than did her rival, who’s more known for her serving vulnerabilities than powers.

Serena had trouble getting traction in the match right from the get-go. She was broken in the third game, after which Wozniacki recuperated from a 0-30 deficit with a second serve ace and overhead winner to get to 30-all. Another service winner, this one to the backhand, and a backhand placement earned her the hold. You know something is a little different when Wozniacki hits a second-serve ace. The shot was a testament to the resolve and determination with which she came fully equipped tonight.

Wozniacki then survived a game of multiple deuces to hold for 5-3, weathering a typical storm of baseline blasts—and errors—from Serena. After a Serena hold, Wozniacki served out the set with aplomb, delivering a service winner to end it.

Wozniacki broke in the third game of the second set as well, and while Serena began to find the range with her groundstrokes and the corners of the boxes with her serve, Wozniacki wouldn’t loosen her grip on the match—not until she was so far in front that only a tennis miracle would alter the outcome.

Serena has delivered miracles before, and it looked as if she might pull it off again. But there was no back-up in Wozniacki tonight.