Following the disastrous defeat at home to Brentford on the weekend, Mauricio Pochettino's side delivered a confidence-boosting win over the Championship team to book their spot in the quarter-finals of the cup, in the pursuit of a very welcome piece of silverware.
Chelsea made a fast start in response to that dismal derby, Nicolas Jackson winning a corner with a blocked effort on goal after barely 15 seconds. There were further opening for Reece James and a solid appeal for a penalty on Sterling waved away as the Blues took charge.
The hosts led when Conor Gallagher's cross was flapped at by goalkeeper Leopold Wahlstedt, allowing Badiashile to adjust his body and slot home from 10 yards out.
Blackburn found their feet and came back into things with their own harshly-rejected shout for a spot-kick, this one for a Gallagher handball.
Chelsea started fast once more in the second half, James firing wide in the opening seconds before his defence fell asleep at a simple long ball over the top. Harry Leonard spun inside but tamely fired his side's best chance wide.
It was Rovers' dozing that led to the second goal however as Callum Brittain gave the ball away inside his own half, before Sterling was fed to deliver the clinical blow that put Chelsea in the next round.