The American, whose only singles title at Roland Garros came in 2002, entered the tournament on a 24-match winning streak and showed no signs of taking her foot off the gas as she steamrollered the 83rd-ranked Georgian in 51 minutes.
Williams, chasing a 16th grand slam title, looked ominously athletic as she wrapped up victory in an unseasonably chilly Paris, putting to bed the memory of her only first round loss at a grand slam, last year's exit to France's Virgine Razzano.
After her opponent sent a forehand wide on match point, she waved to all four sides of the stadium before embarking on the greater test of conducting the courtside interview in French.
"I am very happy, this year I am playing very well, but you never know," the 31-year-old said.
"I was definitely nervous. I have to say I'm always a little nervous going into first-round matches at slams, but this time I wasn't as nervous as I was previously.
"For the most part, I felt pretty safe and felt good about my game and that if I can just do what I do in practice, I'll be okay."
She was then being quizzed about her French skills.
"I have been speaking French for years and years, but I don't really have a lot of confidence," she added.
"I just had to kind of jump in. Like once I get there and I get warmed up, I know how to say things and what I can speak.
"It's just getting that confidence to speak in French. It's way, way, more nerve-wracking than playing tennis."
Serbian 14th seed Ivanovic, who lifted the Suzanne Lenglen Cup in 2008, breezed through the opening set but dropped serve in the fourth game of the second on a chilly Parisian morning.
Martic levelled the tie with a smash, only for the world number 107 to collapse early in the decider on Court Philippe Chatrier.
Ivanovic showed signs of nerves too as she closed in on victory, losing three games in a row and wasting four match points.
She prevailed on the fifth occasion, though, when Martic sent a backhand long after one hour 50 minutes.
Little-known Puerto Rican player Monica Puig sprang the biggest shock of the opening day, the world number 85 beating 11th seed Nadia Petrova of Russia 3-6 7-5 6-4.
First-round results on Sunday (women):
Urszula Radwanska (Poland) beat 30-Venus Williams (U.S.) 7-6(5) 6-7(4) 6-4
26-Sorana Cirstea (Romania) beat Kiki Bertens (Netherlands) 5-7 7-5 6-2
Galina Voskoboeva (Kazakhstan) beat Grace Min (U.S.) 4-6 6-4 7-5
Virginie Razzano (France) beat Claire Feuerstein (France) 7-6(6) 6-4
Dinah Pfizenmaier (Germany) beat Mandy Minella (Luxembourg) 7-5 6-1
Shelby Rogers (U.S.) beat Irena Pavlovic (France) 6-3 6-4
Monica Puig (Puerto Rico) beat 11-Nadia Petrova (Russia) 3-6 7-5 6-4
Caroline Garcia (France) beat Yuliya Beygelzimer (Ukraine) 6-3 6-4
Mallory Burdette (U.S.) beat Donna Vekic (Croatia) 6-3 6-4
Yulia Putintseva (Kazakhstan) beat Ayumi Morita (Japan) 6-2 6-3
1-Serena Williams (U.S.) beat Anna Tatishvili (Georgia) 6-0 6-1
19-Anastasia Pavlyuchenkova (Russia) beat Andrea Hlavackova (Czech Republic) 4-6 7-6(5) 6-4
14-Ana Ivanovic (Serbia) beat Petra Martic (Croatia) 6-1 3-6 6-3
Petra Cetkovska (Czech Republic) beat Olga Puchkova (Russia) 6-0 6-2
5-Sara Errani (Italy) beat Arantxa Rus (Netherlands) 6-1 6-2