Basel had taken the lead in the 48th minute when Chilean midfielder Marcelo Diaz gave keeper Ciprian Tatarusanu no chance with a delightful curling shot from outside the area after a blunder by Iasmin Latovlevici.
The Swiss club, who beat Chelsea in their opening game, are now third in the group with four points, two behind the Londoners and Schalke 04.
Steaua, winners of the European Cup in 1986, have one point from three matches.
"The result was fair," Steaua coach Laurentiu Reghecampf told DigiSport. "Our team's attitude was very good and we won our first point, that's all that matters.
"I told the players we need to be more aggressive and not make big mistakes because we are up against very strong opponents."
The Romanians enjoyed a lot of possession but Basel created more scoring chances after finding holes in the home team's rearguard.
Basel almost took the lead in the 12th minute when centre back Ivan Ivanov's close-range shot was cleared off the line by defender Daniel Georgievski.
Tatarusanu then did well to parry an effort from Fabian Frei while Lukasz Szukala's brilliant block denied Mohamed Salah.
Behrang Safari also went close for the visitors when his shot hit the side-netting.
At the other end Cristian Tanase's shot from 25 metres was comfortably saved by Yann Sommer.
Valentin Stocker then hit the bar for Basel with a deflected shot five minutes before the break.
Steaua refused to give up and earned their reward when Brazilian striker Tatu, who came on as a second-half substitute, scored to delight the home fans.
"It was a game we wanted to win and it's a pity we missed our opportunities in the first half," said Basel coach Murat Yakin.
"But this is a positive result for us because we played against tough opponents today. Romania has quality players and a very strong championship."