The former world and European champions have 14 points from seven matches, four behind leaders Serbia. The two teams now meet in Belgrade on Wednesday, with only the Group Seven winners certain to qualify.
France wasted several opportunities before captain Thierry Henry put them ahead three minutes into the second half, sending a William Gallas header into the net from close range.
"To create as many chances, to show so many things and not manage to score, that's bad luck," said France coach Raymond Domenech, who has faced heavy criticism in the local media since his side flopped at Euro 2008 and might struggle to keep his job in case of a bad result in Serbia.
"I'm not blaming anyone," he added. "What we did was great. We were just not lucky. Our fate is in our own hands. We can still qualify, no matter how."
Domenech's Romanian counterpart was not complaining.
"It's an important result for us against a powerful side with plenty of individual talent," said Razvan Lucescu, whose team stay second from bottom in the group.
"France had more chances but we kept fighting and never accepted the idea that we might lose," he added. "A match like this one gives us confidence and will help us grow".
