The Swiss side were poised to go level at the top of Group E with both Chelsea and Schalke when Marcelo Diaz scored three minutes after half-time.
But Murat Yakin's men were unable to build on the Chilean's goal and with two minutes left Leandro Tatu earned the Romanians a point.
It is unlikely to alter their chances of going through, even to the Europa League, but for Basel it means that Chelsea and Schalke remain in control of the top two spots.
Knowing they had no points, Steaua certainly looked up for the fight early on and it took just 22 seconds for them to create a chance, with Iasmin Latovlevici having a shot charged down by Fabian Schar.
That did not lead to a sustained spell of pressure, though, and the next opportunity fell to Basel, with Ivan Ivanov having a strike hacked clear by Daniel Georgievski. From the subsequent corner Fabian Frei found Diaz but his header was repelled by Ciprian Tatarusana.
Steaua responded as Nicolae Stanciu drove over and Cristian Tanase worked Yann Sommer from range, but Basel continued to give as good as they got and Behrang Safari went close from six yards.
Tanase then drew a fine stop from Sommer and Stanciu wasted a free header as the pressure continued, before Basel went the closest of all as Valentin Stocker thumped the bar from 20 yards.
That would prove to be the pre-cursor to the opener as Diaz found the net with 48 minutes on the clock.
Marco Streller set him up, passing in from the right, but the rest was down to Diaz who curled in from 25 yards.
Serey Die and Stocker then had efforts charged down as Steaua offered nothing in response, before they eventually stirred on the hour when Latovlevici hit wide.
It was Basel who looked the likelier to get the next, though, and Frei almost got it with a low strike which skipped away from goal, before Sommer produced a fine parry to deny Lucian Filip as Steaua's plight dawned on them.
That continued with Alexandru Bourceanu going close and Latovlevici having a shot charged down and their rallying produced an 88th-minute reward.
Adrian Popa swung the ball across and Tatu was well set to smash in from the penalty spot and earn his side a point.