The US Open runner-up burst out of the blocks and wrapped up the first set in just 25 minutes under the blazing sun on Court Philippe Chatrier.
Kudryavtseva, the world number 78, stopped the rot by winning the eighth game and had the audacity to break for a 2-1 lead before Wozniacki, 19, restored order, making the decisive break in game eight for a 5-3 lead.
She sealed the win after an hour and nine minutes when the Russian netted a backhand service return. She now plays Tathiana Garbin for a place in the third round after the Italian recovered from dropping the first set to beat Germany's Kristina Barrois 1-6 7-6(7) 6-3.
World number one Serena Williams recovered from a stuttering start to beat Swiss Stefanie Voegele 7-6(2) 6-2 and book her place in the second round.
The 2002 Paris champion stooped and clenched her fist as she saw off three break points en route to a 7-2 win in the first set tiebreak after struggling to break her opponent's serve.
World number 76 Voegele, backed by the centre court crowd, lost her composure in the second set as Serena ended the contest with a routine volley after an hour and 21 minutes.
She will next face Julia Georges, after the German beat Hungary's Melinda Czink 6-2 6-3.
The Russian had won just two matches in four tournaments coming into Roland Garros but seemed determined to set the record straight with a emphatic win in just 69 minutes.
Dementieva, who lost to Anastasia Myskina in the 2004 final in Paris, will meet tricky Spaniard Anabel Medina Garrigues, after she beat American Melanie Oudin 7-5 6-2.
Ana Ivanovic continued her better run of form with a solid 6-3 6-3 win over talented youngster Chang Kei-Chen of Taipee.
The 22-year-old Serb, champion in Paris just two years ago, has plummeted to world number 41 after a run of woeful form.
But victories over Dementieva and Nadia Petrova in Rome earlier this month suggests that Ivanovic may be rediscovering some of the form that took her to world number one in 2008.
And next up for Ivanovic will be 28th seed Alisa Kleybanova, after the Russian beat Mariya Koryttseva 6-2 6-1.
China's Li Na used her experience to fend off teenage French wildcard Kristina Mladenovic 7-5 6-3 on a baking hot Suzanne Lenglen court.
The 28-year-old Li, who became the first Chinese player to attain a top 10 singles rankings after reaching the Australian Open semi-finals in January, dropped serve twice in the opening set but replied with three breaks of her own.
Mladenovic, rewarded with a main draw spot after winning the girls singles title last year, was bothered by a back strain in the second set but still produced a few scorching winners to keep the home fans interested and Li on her toes.
Li, the 11th seed, went 0-40 down when serving for the match but responded with five straight points to move through.
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