The third seed was never really threatened but let her guard down too many times for comfort against a willing opponent on a sun-drenched court Philippe Chatrier.
Halep, the top name in action in the women's draw on day one, broke twice to open a 4-1 lead, only for Rodina to hit back and level at 5-5.
The Russian let another service game slip from her hands with a double fault and Halep clinched the set with a drop shot.
She broke decisively in the ninth game of the second set and wrapped it up with an ace.
"It was a tough match, because it was first round and it is always difficult to start the tournament," Halep, who lost to Russian Maria Sharapova in last year's final, told a news conference.
"But she played well. I (made) easy mistakes, but it's normal and I accept that. Next round I will be better, for sure."
While most of the focus is on Sharapova and top seed Serena Williams, Halep said she is better-equipped to challenge the big two this year, having grown as a player.
"I'm stronger, I think, mentally, and I'm stronger, as well, on court. I feel stronger -- I have a stronger body and I have improved a lot in my game. Serve is better, so I feel more confident now with my game," she explained.
She will next face Croatian Mirjana Lucic-Baroni, who beat her in the third round of the U.S. Open last year.
It will be a different story on the Paris clay, though, according to Halep.
"I have no good memories from that match, because I didn't play my match. I just was blocked and I couldn't hit the ball," Halep said. "But now I feel much better. I feel that I have my chance.
(Reporting by Julien Pretot; Editing by Martyn Herman)
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