A strong Blues line-up numbering five first-team regulars including Pedro and Nemanja Matic laboured against their League One opponents despite twice claiming two-goal leads as Michy Batshuayi scored either side of Victor Moses’ goal.
Peter Hartley and Ellis Harrison, who fired the only goal of the second-half, gave Rovers hope of a famous upset against the Premier League giants and on more than one occasion they threatened an equaliser.
But Eden Hazard and Oscar came on to reinforce Chelsea and the visitors eventually ran out of steam having given everything in pursuit of a prized scalp.
Batshuayi celebrated his full debut after impressive cameos against West Ham and Watford with a brace, although the £33million summer signing from Marseille was also wasteful.
Antonio Conte opted against promoting Diego Costa from a heavyweight bench and the Brazilian-born marksman seems certain to be restored to the starting XI at Batshuayi’s expense for Saturday’s Premier League clash with Burnley.
It was the Belgium striker who forced an early save from keeper Steve Mildenhall at the near post and during a scramble from the ensuing corner, captain Branislav Ivanovic almost bundled the ball home.
The Belgian should have propelled Chelsea ahead in the 19th minute after smart passes from Cesc Fabregas – who was making his first start of the season – and Ruben Loftus-Cheek sent him goalbound.
Having looked up to pick his spot, Batshuayi then scuffed a tame shot across the face of goal.
Loftus-Cheek struck the left post after marauding through Rovers’ defence and the 20-year-old was involved when Chelsea stormed ahead just before the half-hour mark, finding Matic, who pulled back for Batshuayi to drive home.
In a heartbeat the lead became 2-0 after Pedro – who also played a key role in the build up to the opener – crossed to Cesar Azpilicueta and when he stabbed the ball back to Moses who was well-positioned to finish.
Chelsea’s defence responded to the commanding position by losing focus, enabling Hartley to nod in Chris Lines’ free-kick before Harrison headed a great chance wide moments later.
Any panic was swiftly quelled, however, when Fabregas picked out the dangerous Loftus-Cheek who rounded his marker and teed up a powerful finish for Batshuayi.
An entertaining match continued its unpredictable path into the second half when Matty Taylor raced in on goal only to be brought down by Pedro, who was shown a yellow card.
Referee Keith Stroud also pointed to the spot and Harrison made no mistake, while long range attempts from Harrison and Jermaine Easter cranked up the pressure on Chelsea.
On came John Terry for injured debutant Ola Aina, joining Hazard among the reinforcements summoned from the bench by Conte.
Rovers were tiring and Chelsea were now back in control, knowing they had weathered the storm.