Former world number one Williams, who has won seven Grand Slam events but is rebuilding her game and her life after being diagnosed with an autoimmune disease, got into her stride only at the end of a one-sided match.
Radwanska dominated the court and went 5-1 up in the second set before Williams began to fight back.
The American, watched by sister Serena who made a shock first-round exit on Tuesday, won the next game to love then broke for 5-3.
With light rain beginning to fall on the Philippe Chatrier court, Williams was one point from winning the next game but Radwanska got to matchpoint when she scooped the ball over the American's head, leaving her standing.
A forehand mistake by Williams finished off the match in exactly an hour.
World number one
The Belarusian, who was made to work hard in the first round against Italian Alberta Brianti, needed just 55 minutes to crush the world number 198.
Azarenka lost the first game but went on to reel off 11 in a row as she overpowered her 20-year-old opponent.
"I didn't really know my opponent today, and it took me the first few games to kind of understand what she does and what kind of game she plays to adjust a little bit," Azarenka told reporters. "But after a few games I started to find my better rhythm. Definitely played much better today."
Next up for the Australian Open champion will be Canada's
"She's a tricky opponent," Azarenka said of her next test. "I definitely have to come up with good tennis in my next round."
US Open champion
"I thought it was a good, solid match," the Australian sixth seed said. "I was happy with the winners that I hit. I thought you could always hit a few less errors... Overall, I'm quite happy with it."
The eighth-seeded Bartoli, a semi-finalist at Roland Garros last year, served 14 double faults as she crashed out in two hours and 12 minutes against the world number 50.
Bartoli came from a break down in the second set to level the tie but she never looked comfortable on Court Suzanne Lenglen under threatening skies.
Martic served an ace on her second match point to set up a third-round meeting with Spanish 29th seed
"Last year probably was the hardest one for me ever," said the Serbian after beating Israeli
"To come back here this year and play better and go through to the third round, it means a lot. I really hope I can even go further."