Meanwhile, Samantha Stosur of Australia won her second title on the women's WTA Tour circuit on Sunday.
Sharapova later pulled out of this week's WTA tournament in Montreal after suffering the injury to her foot.
BBC: Kim Clijsters beats Maria Sharapova in Cincinnati final
Wozniacki will look to return to winning ways at this month's WTA tournament in Qatar, where last year she lost in the final.
Lindsay Davenport will miss next week's Los Angeles Classic with the same knee injury that saw her withdraw from this week's WTA Tour event.
The reigning French Open champion will also skip next week's WTA tournament in Sydney to recover in time for the start of the Australian Open on January 14.
CNN: Injured Sharapova pulls out of season opening tournament
Meanwhile, on the women's WTA Tour, top seed Elena Dementieva of Russia fought back from a set down to beat Czech Lucie Safarov in the final of the Open GDF Suez tournament in Paris.
Defending champion Vera Zvonareva of Russia is top seed for this week's other WTA Tour event, the Pattaya Open in Thailand.
It was a schism that led to the formation of the establishment-approved Women's Tennis Association in 1973 -- which will be marked this year by the WTA's "40 Love" commemorative campaign, a celebration of four decades of progress in the women's game.
"It was definitely a match I had to be really focused for, because she's a really good player, " Serena told the WTA Tour's official Web site.
Since mid-summer she has claimed the Wimbledon and U.S. Open titles before coming out on top in the WTA Tour's finale, despite having not played since her Flushing Meadows triumph.
"I didn't do much wrong today besides the first few games on my serve, " Ivanovic, 24, told the WTA's website.
Elsewhere former women's world number one Ana Ivanovic lost to 21-year-old German wild-card Mona Barthel in the first round of Stuttgart's clay-court WTA tournament.
"I was happy to have a quick match, " Radwanska was quoted as saying on the WTA Tour's official website.
The 26-year-old's last win on the WTA Tour was in Stanford that year.
The 23-year-old won her first grand slam title in Melbourne last January before going on to top the rankings and become the WTA Tour's leading prize money winner for 2012.
But she returned with a bang in 2012, winning Wimbledon, Olympic Gold, the U.S. Open and the WTA Championships, while she has also triumphed in 59 of her past 62 matches.
"First of all we still have to get to that point, and then we can discuss it further, " Sharapova told the WTA Tour's website when asked about a possible meeting with Williams.
"I had never faced my opponent before, and when she has time she really goes for her shots, so I wanted to get her on the move, " the 25-year-old Sharapova said on the WTA Tour's website.
"After a really long match in my previous round, I was just trying to be really aggressive today and not spend as much time as I did the other day, " Sharapova told the WTA Tour's official website.
"In the beginning I was trying to shoot as hard as I could, and I was making a lot of mistakes -- so I knew I had to start putting the ball back in the court more, " Li told the WTA Tour's official website.
"I've never felt this fit, this strong, this happy to play -- and I think I can be even better, " fourth-ranked Williams told the WTA Tour's website after her 7-5 6-3 win over the 20-year-old Vandeweghe, who made it into the main draw as a lucky loser from qualifying.
The Wimbledon women's singles seedings reflect the current WTA Tour rankings, with world number one Sharapova selected top.
Watson's previous best effort at a WTA event came in Eastbourne last year when she made it through qualifying to reach the last 16.
Meanwhile, on the WTA Tour, China's Li Na reached the quarterfinals of the Stuttgart tournament with a straight sets win in her first match of the clay court season.
The UK telecoms firm has been investing heavily in its sports offering, including buying the rights to show live Aviva Premiership rugby union matches and a deal with the WTA to show women's tennis.
On the women's tour, Mona Barthel won her second WTA title after upsetting top seed Sara Errani in the final of the Paris Open .
Williams finished 2012 in ominous form, winning Wimbledon, the U.S. Open and the season-ending WTA Championship, as well as taking double Olympic gold in London.
Safarova didn't leave Charleston empty-handed, however, securing her first WTA doubles title alongside Russia's Anastasia Pavlyuchenkova as they beat Spaniards Anabel Medina Garrigues and Yaroslava Shvedova 5-7 6-4 10-6.
Ever since the pair appeared on the WTA Tour in the 1990's -- complete with multi-colored braided hair -- they have been setting, and sometimes destroying, trends with their extroverted dress sense.
With Heather Watson and Laura Robson poised to leave the juniors and make an impact on the WTA Tour, the women's game is in better shape than at any other time in the last decade.
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