The 26th seed, who won the 2009 title at Roland Garros, won 6-1 6-2 with a display of relentless consistency to which the frequently wayward Radwanska had no response.
Despite her lithe frame, the Pole looked ill at ease on the clay and lacked the all-round aggression of her Russian opponent who broke four times to take the first set in 30 minutes.
Kuznetsova continued to punish the out-of-sorts Radwanska to book her place in the fourth round.
The Russian had beaten Radwanska in their previous five meetings and extended her already impressive head-to-head record to 10-3.
"It was very important to dictate, and I think I have done it well," the world number 28 told reporters. "I fought for every ball against her.
"When you look at her when she plays, she doesn't have some great shots, like overpower. She moves well, she reads the game unbelievably, and she plays on the counterattack very well."
Radwanska, who had enjoyed an impressive start to 2012 winning titles in Dubai, Miami and Brussels, was sluggish out of the blocks, losing her first three service games while struggling to punish her opponent's occasional errors.
She had chances to claw her way back into the match but converted only two of 10 break point opportunities
"She was just playing too good today," Radwanska told reporters. "She's a great player on clay, that's for sure.
"She was just playing very aggressive on both sides. I think she just had answer for everything that I was trying to do."
The Australian Open champion from Belarus came out on top of a hard-hitting contest after 86 minutes just before dusk on Court One.
Azarenka, who will retain her world number one spot if she reaches the quarter-finals on the Parisian clay, broke decisively in the ninth game to claim the opening set and she was barely bothered in the second.
She will next face Slovakian 15th seed Dominika Cibulkova, who beat Spain's Maria Jose Martinez Sanchez 6-2 6-1.
On Thursday, she refused to hang around longer than necessary on a cold and blustery Suzanne Lenglen Court and whipped winners at will to end 84th-ranked Morita's challenge as the clock ticked over the hour mark.
Morita's backhand error on match point allowed Sharapova to become the final singles player to reach the third round, where she will face China's Peng Shuai.
"It was a pretty long day yesterday. I feel like I warmed up like 20 times for this match," said Sharapova.
"It was one of those days where you just want to get on the court and then, you're (hanging around) all day, sitting, waiting around, eating, sleeping. It's like a good way to put someone into retirement.
"It was nice to get out there today and finish it."
Sharapova now platys Shuai Peng in the third round.
Former champion Ana Ivanovic went out, beaten 1-6 7-5 6-3 by Italian claycourt specialist Sara Errani in an erratic third-round display.
Ivanovic, the Roland Garros winner in 2008 and a former world number one, started well on the Suzanne Lenglen showcourt but her game deteriorated and, at one stage in the final set, she hit three double faults in a row.
Errani, seeded 21st while Ivanovic was number 13, served well and won points at the net before taking victory on her second matchpoint when the Serbian put a forehand wide.
World number 14 Ivanovic took full responsibility for her loss, saying: "I was making some unforced errors when I was too flat-footed.
"In the third set I was creating a lot of opportunities and missing a lot of easy, easy finishing balls and that is something I am not happy about.
"Some double faults really came out of nowhere. Maybe I wasn't using my legs enough."
Errani, who said her coach had told her to play to Ivanovic's backhand as much as possible, was delighted with the win.
"It is one of my best," she told a news conference. "She is not top 10 now but she has always been among the top players, so I am very happy."
Australian sixth seed Stosur, the runner-up at Roland Garros two years ago, wasted little time and energy seeing off Petrova, who turns 30 next week, on Court One.
Though Stosur was a model of efficiency, the match had few highlights and one front-row spectator could be seen nodding off in his seat.
The two women had played seven times before, with Petrov winning five, so the 28-year-old Stosur had worked out her game plan thoroughly.
"We have had some pretty long matches in the past and she had got the better of me a lot of times," said Stosur, who beat Petrov at the same stage in New York last September in the longest women's match at the U.S. Open since the introduction of the tiebreak.
"I knew what I wanted to do, I stuck to the plan and was able to...execute it very well," added Stosur. "I thought today was very, very solid and a good match."
The 28-year-old Stosur now faces American Sloane Stephens, who beat Mathilde Johansson, the last Frenchwoman left in the draw, 6-3 6-2.
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