The win, Unirea's first at home in European competition, put the Romanians onto eight points in Group G, two more than third-placed Stuttgart. The two sides meet in Germany in the final group game on December 9 with Unirea needing a point to progress.
Sevilla, who had already qualified and have 10 points, dominated the opening half with strikers Frederic Kanoute and Alvaro Negredo both hitting the post.
But against the run of play Champions League debutants Unirea went ahead seconds before halftime when Dragutinovic, under no pressure, headed a lofted Pablo Brandan free-kick from near halfway into his own net.
"We are extremely happy, we did not manage a spectacular game but we have managed to win," Unirea defender George Galamaz told public television TVR 1.
"I don't know what happened to us."
The second half was more balanced, but Unirea midfielder Dacian Varga missed three great opportunities to score while Sevilla could not break down the Romanian team's well-organised defence.
Sevilla substitute Luis Fabiano almost levelled in stoppage-time but fired inches over the bar.
While no Unirea player has scored more than once in Champions League, the debutants from rural Romania have taken benefitted from three own goals by their opponents in this competition.
Coach Dan Petrescu, a former Chelsea defender, was disappointed Rangers failed to snatch a point from Stuttgart after the German side won 2-0 in the other group game.
"When you win against a team like Sevilla and you have eight points, you would like to have been already qualified," Petrescu told TVR 1.
"But I guess every match in this competition has to be the match of our lives. You get tired at some point."
Key Unirea defender Vasile Maftei will be suspended for the final group game after a late booking.
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