It is 18 months since the 22-year-old last played in England's top flight as his promising Chelsea career petered out, but, after a year on loan in Germany at Bayer Leverkusen, he was back with a bang.
He was Forest's brightest spark following his deadline-day move from Stamford Bridge and brought his side level in the second half after Zeki Amdouni had put the Clarets ahead.
"Any new player will want to hit the ground running and show what they can do," Forest manager Steve Cooper said.
"I know (Hudson-Odoi) he has quality and he can have moments of brilliance. We need that at times in the Premier League and we got one tonight.
"It is a brilliant goal. We have had a few of those scored against us but we haven't had too many of them for us. Beautiful strike, beautiful technique, great for him, great for the team. He will feel good about that."
The point for Burnley enabled them to get their campaign up and running after three successive defeats but they will feel they should have taken all three after a goal by Lyle Foster was controversially disallowed in the 76th minute. Adding to their ire the frustrated Foster was sent off in added time for elbowing Ryan Yates.
Hudson-Odoi has been reunited with Cooper, who managed the winger when he was part of the England U17 World Cup-winning squad.
The equaliser came after Taiwo Awoniyi laid it off to Hudson-Odoi, who cut inside and sent a 20-yard curling effort in off a post, with the goal surviving a VAR check.
Burnley were not so lucky with VAR. It ruled Sander Berge used his hand to get past Scott McKenna and tee up Foster to score. The video review process then caught Foster's rogue elbow with referee Robert Jones again ruling against the visitors after checking the pitchside monitor.
"I have a decent business brain and a decent coaching brain, but when it comes to the laws and legalities I switch off," Burnley boss Vincent Kompany said.
"It's not something I want to discuss too much because I can't change it now.
"I thought today was at least even and depending what side of the fence you sit on you probably think you deserved three points."