But from then on it was one-way traffic with Sharapova devouring the Argentine's serve and despatching winners with ruthless aplomb. She won the match on her first match point when the Argentine hit long.
Sharapova will play Samantha Stosur of Australia, the 19th seed, in the last 16.
Following the exit of Serena Williams and Li Na, third seeded Polish player Radwanska could not benefit as she tumbled out courtesy of an indifferent performance. The former Wimbledon finalist lost 6-4 6-4 to Ajla Tomljanovic.
21 year old, world number 72 Tomljanovic from Croatia dictated play with a beguiling blend of power and feel which didn't allow her opponent to get a grip.
It is the first time since tennis turned professional in 1968 that the top three seeds in the women's draw have failed to qualify for the fourth round of a grand slam.
Australian Open champion Li was knocked out in the first round by France's 21-year-old Kristina Mladenovic, ranked 103, and Williams was beaten in the second round by 20-year-old Garbine Muguruza of Spain.
"After seeing the two first seeds go out, you kind of feel you can do this, too," said the 21-year-old Tomljanovic.
"I grew up with these girls who are beating them. I went into the stadium for the first time, and she kind of feels like home there, because she's been there a lot more than I have.
"I went out there, and inside I really thought I could win. I think that showed and it is why I won."
Poland's Radwanska reached the Australian Open semi-finals in January but played poorly on Friday, causing the world No. 72 few problems.
"This year I've been in a few situations when I did not execute and this time I wanted to make it right," the Florida-based Tomljanovic said.
Tomljanovic unsettled Radwanska with sliced backhands and opened a 5-1 lead after breaking serve twice. The Pole pulled one back for 5-3 with a forehand winner but a lob that sailed long handed Tomljanovic the opening set.
She kept her composure, mixing it up with drop shots to break in the first game of the second set and holding serve throughout.
She ended the contest on the first match point when Radwanska hit long in yet another unforced error.
Radwanska summed up the defeat by saying: "I don't think it was my day today."
Tomljanovic was joined in the fourth round by former finalist Samantha Stosur of Australia, who beat Slovakian ninth seed Dominika Cibulkova 6-4 6-4.
"I'm very happy with my tennis, today was definitely the toughest match. I am very happy to get through in straight sets," said the 2011 U.S. Open champion who now has a 5-0 record against Cibulkova.
American teenager Taylor Townsend's run at the French Open was stopped by Carla Suarez Navarro of Spain 6-2 6-2 in the third round on Friday and Navarro will next face Tomljanovic.
Townsend took time to settle, losing her first two service games to go 4-0 down to the claycourt specialist from Spain who won the Estoril title last month.
The 18-year-old broke back for 4-2 and caused Suarez Navarro problems with her clean hitting and working of the angles, but her desire to shoot for the lines and blaze away from unpromising positions repeatedly worked against her best interests.
She fired wide and long, and netted a volley to give her opponent set point before hitting a return of serve long to go one set down after 24 minutes.
The American wasted four break points in the fourth game of the second set by not playing the percentage game and was broken to love in the next game.
Suarez Navarro's first serve was the match's most potent weapon and Townsend returned long to give the Spaniard victory on her third match point.
Results from the French Open women's round three matches
Ajla Tomljanovic (Croatia) beat 3-Agnieszka Radwanska (Poland) 6-4 6-4
7-Maria Sharapova (Russia) beat Paula Ormaechea (Argentina) 6-0 6-0
19-Samantha Stosur (Australia) beat 9-Dominika Cibulkova (Slovakia) 6-4 6-4
14-Carla Suarez Navarro (Spain) beat Taylor Townsend (U.S.) 6-2 6-2
18-Eugenie Bouchard (Canada) beat Johanna Larsson (Sweden) 7-5 6-4
Pauline Parmentier (France) beat Mona Barthel (Germany) 1-6 6-1 7-5
Garbine Muguruza Blanco (Spain) beat Anna Karolina Schmiedlova (Slovakia) 6-2 6-4
TWEET: Third seed Radwanska lost in R3. For first time in Open era HISTORY the top-3 women's singles fail to reach L16. #RG14 (1/2)
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- Samantha Stosur
- Taylor Townsend
- Carla Suarez Navarro
- Serena Williams