Thomas Tuchel, whose side came into this match on the back of a run of just one win in their previous five matches across all competitions, opted for a much-changed lineup with several young players in the XI following a minor COVID outbreak at the club.
His side will next face Tottenham Hotspur and former Blues boss Antonio Conte in the two-legged semifinal, commencing the week of Jan. 3.
After a cagey opening, Chelsea began to dominate as the first half wore on without creating any real chances.
Chelsea debutant Harvey Vale could have opened the scoring with a header, while stand-in goalkeeper Kepa Arrizabalaga produced several decent saves to keep Brentford at bay.
And it arrived with 10 minutes remaining as Jansson sliced James' cross into his own net to hand Chelsea the victory.
Five minutes later, Jorginho made the result safe as he coolly slotted home from the penalty spot after Brentford goalkeeper Alvaro Fernandez had down Pulisic.
Tuchel has the chance to fight for his third major trophy at Chelsea following the Champions League and UEFA Super Cup victories.
"It was a brilliant result given the circumstances, the line-up and the situation," Tuchel said. "Some of the guys who played were youngsters and the other guys had not many minutes, but they stepped up in a tough place."