Hudson-Odoi's sixth-minute opener at Stamford Bridge got the Blues off to a flyer and Ross Barkley doubled the lead before half-time.
It was a welcome win for Chelsea, which had lost three of its past four home games in the Premier League.
Second-tier side Forest was unable to lay a glove on the eight-time winner.
Chelsea, which also knocked Forest out at this stage last year, opened the scoring early on.
Quick build-up play led to Hudson-Odoi being fed on the right, with the England international cutting inside and firing a low finish through the legs of Yuri Ribeiro to beat Jordan Smith.
The Championship side was awarded a penalty in the 22nd minute when Alex Mighten was bundled over by Fikayo Tomori, but a VAR check showed the lively 17 year-old was offside.
Chelsea then doubled the lead, Smith parrying Hudson-Odoi's powerful drive, with Barkley arriving late to tuck home the rebound at the back post.
A third almost arrived on the hour-mark, Smith making a great save to tip Barkley's header onto the post after Hudson-Odoi opened up the Forest defence once more.
Ryan Yates then had a close-range header ruled out for offside at the other end.
Teenage defender Tariq Lamptey was given another cameo off the bench in the closing stages, when Chelsea almost got a third through Hudson-Odoi only for Smith to push his try wide with the final act of the game.