Hudson-Odoi's sixth-minute opener at Stamford Bridge on Sunday got the Blues off to a flyer and Ross Barkley doubled their lead before half-time.
It was a welcome win for Chelsea, who had lost three of their past four home games in the Premier League.
Second-tier side Forest, who were much changed, were unable to lay a glove on the eight-time winners.
Chelsea, who 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 were 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 their 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.
What does it mean? Chelsea still need to spend
Forest are going well in the Championship's top four, but Lampard may have expected more from some squad players who were given a chance to shine. It has been reported that Jadon Sancho, Wilfried Zaha and Moussa Dembele are among the Blues' January targets after their transfer ban was reduced on appeal.
Hudson-Odoi makes case for starts
One player who did impress was Hudson-Odoi, the 19-year-old providing a regular threat from the right wing.
He has only started three games in the Premier League this season but could have persuaded Lampard to trust him in the XI more often in the coming weeks.
11 - Callum Hudson-Odoi has been directly involved in 11 goals in his 13 starts at Stamford Bridge for Chelsea in all competitions, scoring five and assisting six. Alternative. pic.twitter.com/eDH7g1lbg1— OptaJoe (@OptaJoe) January 5, 2020
Batshuayi draws a blank
Michy Batshuayi offered very little in a rare start up front and, with the Blues said to be hunting attacking reinforcements, he could not afford to have a quiet day.
Missing a good chance from another fine Hudson-Odoi cross epitomised his lack of confidence in front of goal. Batshuayi had scored 10 times for Chelsea in seven domestic cup games against lower-league opposition.
Burnley visit Stamford Bridge in the Premier League on Saturday, when Forest go to Reading in the Championship.