While Lewis Grabban scored the only goal in the 83rd minute, Cook's efforts prevented Mikel Arteta's team from having a single shot on target in a game televised live on ITV.
Cook wrote on Twitter afterwards: "Debuts don't get much better than that. Unbelievable win and a great atmosphere. It was special being out there."
The 30-year-old signed for Forest last Wednesday on a two-and-a-half-year deal after spending more than a decade with fellow Championship outfit AFC Bournemouth.
He told Forest's website: "I thought it was the perfect time in my career to make this move. You can see the progress the club is making and I'm excited for the new challenge."
Cook initially joined the Cherries on loan in 2011, before making the move permanent the following year, and went on to make 389 appearances for the Dorset club, scoring 20 goals.
He helped Bournemouth win promotion from League One in his first full season and was an ever-present in the club's Championship title triumph of 2014-15.
The former William Parker schoolboy later missed just 21 of Bournemouth's 190 games during their five-year stay in the Premier League.
But a knee injury kept him out at the start of this season and he featured only four times in all competitions for Scott Parker's Bournemouth in 2021-22.
Head coach Steve Cooper told Forest's website: "As soon as we knew of his possible availability, we went for it as much as we could to see if we could make it happen.
"We feel this is an important signing as Steve is a fantastic player and brings a good level of experience, both in the Championship and the level above."
Although now departed, Bournemouth have confirmed that Cook, who was appointed club captain last term, will receive a testimonial at the end of the current campaign.
Cherries chief executive Neill Blake added: "Steve is an AFC Bournemouth legend. He has been a key part of this club's success over the last 10 years and has played a pivotal role in some of the most memorable moments in our history."