Serdar Tasci fired Stuttgart ahead after five minutes but the Romanian champions equalised three minutes after halftime when midfielder Dacian Varga curled a shot past keeper Jens Lehmann.
"This is the most important goal of my career so far," Varga told public television TVR 1.
"We did not make fools of ourselves. We played a good match and if we were more careful, we would have managed to get all the three points."
The Romanian side enjoyed good spells of possession but carved out few opportunities, the best of which fell to midfielder Tiberiu Balan who blasted over Varga's fine cross.
Stuttgart also had their chances but Romanian striker Ciprian Marica and midfielder Timo Gebhart failed to convert from close range.
"(Unirea) were disciplined and worked hard and I feel bad for our poor game today and I feel bad for missing a big chance to score in the second half," Marica told TVR 1.
"Now it is clear that Sevilla are the group favourites."
The German side, playing in their 150th European game, have now won only one of their last eight Champions League away games.
Unirea coach Dan Petrescu, who was in the Chelsea squad that beat Stuttgart 1-0 to win the 1998 UEFA Cup Winners' Cup, said he was not satisfied with the result.
"I told the boys at halftime that if we increase pressure we will score," he told a news conference. "With a bit more inspiration we could have won. I actually regret that we lost two points today."
