Azarenka, 10 years younger and 104 ranking places higher than her opponent, was in patchy form, giving away breaks with double faults and errors and screaming in anguish at her own mistakes.
After losing the first-set tiebreak 6-8, the Belarusian world number one found herself five points from defeat as she was 0-4 and break point down in the second set.
However, just when it seemed that Brianti was on the verge of winning her first match at Roland Garros in five appearances, Azarenka put the Italian under pressure in a long fifth game.
The Australian Open champion clawed her way back, then broke to love in the eighth game of the third set to take victory in two hours and 16 minutes.
Told at her news conference that she had made 60 unforced errors, Azarenka said: "I think that says it all. Bad days happen. Unfortunately today I had way more mistakes than I usually do but it happens sometimes.
"You can sit and look for excuses, what happened or whatever. I just try to not approach it that way. Just try to think what can I do better next time?
"Tomorrow is the day that I have to work. It's a new opportunity. That's it."
The 22-year-old Belarussian said she was no longer troubled by a right shoulder injury which forced her to pull out of the Italian Open nearly two weeks ago.
"It's much better, the pain went away," she said. "But I didn't have much time to prepare."
China's Li, who became the first player from an Asian nation to take a grand slam singles title when she won at Roland Garros last year, was never challenged by the 43rd-ranked Cirstea.
In front of stands that were barely half-full on Philippe Chatrier Court, Li, who played with her right knee strapped up, allowed the Romanian few chances and few points.
Li said she was still upset about losing to Russian Maria Sharapova in a dramatic and rain-interrupted Italian Open final eight days ago.
"After that final I told everyone: 'Don't talk to me about tennis for three days'," the world number seven said.
Radwanska set up a second-round meeting with American former world number one Venus Williams, who is finding her feet in tennis again after being diagnosed with an autoimmune disease.
Jovanovski tried to fight back at the end of the first set on Court One, saving three set points on her own serve before putting a backhand into the net.
The second set was even more one-sided, with Pole Radwanska allowing her rival a single point in the first three games.
Jovanovski produced another backhand error on matchpoint, and her ordeal was over.
French eighth seed