The 18-year-old winger provided assists for both of Alvaro Morata's second-half goals, which turned the game in Chelsea's favour after Cesc Fabregas missed a first-half penalty for the Blues.
Goal understands Bayern have offered £30 million ($37.5m) for Hudson-Odoi after he was given just 42 minutes of Premier League football by Blues boss Maurizio Sarri in the first half of the season.
The England Under-19 international was one of Chelsea's best players as they secured their place in the fourth round, and when asked whether his display might help him break into Sarri's team, Hudson-Odoi told BeIN Sports: "Hopefully, hopefully. You never know.
"I feel really proud to get opportunity to play and help the team get the win we deserved.
"We played really well, our mentality was strong and we deserved the win that we got. Alvaro did really well getting the goals and we all worked hard as a team and we deserved the win."
Hudson-Odoi was instrumental in the victory over a well-organised Forest side, causing Aitor Karanka's men problems down Chelsea's right flank, where he twice crossed for Morata to score, becoming the first teenager to lay on a goal for the Blues in over a decade.
He also had shots of his own saved by Reds goalkeeper Luke Steele and said Chelsea's persistence typified their mentality under Sarri.
"If we don't score we keep going and keep pushing until we do. I think the way we played showed the desire to keep pushing and the chances were coming," said Hudson-Odoi. "We've just got to be patient sometimes.
"Hopefully today showed why we're doing really well. I'm proud to make history but it's not just about myself it's about the team as well."
Fabregas, who ended the match in tears after playing what is likely to have been his last game for Chelsea, backed Hudson-Odoi to become one of the world's top players.
"He has got everything to make it in world football," Fabregas told the BBC.
"I've told him myself that I'll be disappointed if he doesn't make it because he's one of these talents who you can see can be world-class.
"He's a humble boy, he loves playing football and hopefully he can really make it very soon."