Fernando Torres scored only his second goal in 20 games to send Chelsea into the Europa League quarter-finals.
Trailing 1-0 from last week's first leg in Bucharest, Chelsea took the lead on the night through Juan Mata's close-range finish.
But a scrambled effort from Vlad Chiriches before the break left Rafael Benitez's side needing two goals.
John Terry pulled one back with a header then Torres produced a fine finish before missing a late penalty.
After battling back from two goals down against Manchester United last weekend to secure an FA Cup replay, Chelsea were once again made to do it the hard way.
But last season's Champions League winners showed plenty of patience to grind Steaua down.
Captain Terry has been used sparingly by Benitez because of an ongoing knee problem and he has not started a Premier League game since 2 February.
But his goal set up the comeback before Torres, who only had an FA Cup goal against Brentford to show for his previous 19 games, drove in a third, although there was frustration when he failed to take another chance from the spot.
For all the negativity that has surrounded the Stamford Bridge club this season, they still have two chances for silverware.
But they had to dig deep to make it past Laurentiu Reghecampf's team.
Steaua began with all the confidence of a team 12 points clear at the top of the Romanian table and were unfortunate not to move ahead when Raul Rusescu was played in behind the Chelsea defence only to see his shot turned wide by Petr Cech.
The home side finally imposed themselves when a deep cross from Eden Hazard found John Obi Mikel at the far post and his shot was pushed away by Ciprian Tatarusanu.
Just after the half-hour mark they made the breakthrough when Mata drove forward and swapped passes with Ramires before stabbing a shot with enough power to get through Tatarusanu's legs.
But in first-half added time, the visitors responded when a Iasmin Latovlevici corner was met by Cristian Tanase, whose shot was pushed on to the woodwork by Cech only for Chiriches to squeeze in at the near post to leave Chelsea needing two goals.
Chelsea struggled for fluency at the start of the second period but just before the hour, an unmarked Terry headed in off the post from a Mata free-kick to give the home side hope.
Cech was worked again when he palmed away from Alexandru Chipciu's looping effort.
But Chelsea came up with a decisive third goal when Mata drifted in from the right and his pass was helped on by Hazard before Torres guided a shot into the corner.
With five minutes remaining, Hazard was brought down in the area by Alexandru Bourceanu but Torres hit the bar with the resulting penalty.