"We get put on the back foot and you have to see how the players deal with that," Adams told BBC Radio Norfolk.
"But I thought they were fantastic. They didn't let it affect them and they saw us through."
Forest's Jordan Palmer-Samuels cancelled out Reece Hall-Johnson's goal from the first leg, before City defender Cameron Norman saw red on the hour mark for a second bookable offence.
But the Canaries clung on through extra-time and won 5-4 on penalties, with Hall-Johnson converting the decisive spot-kick in front of a crowd of 9,690 at Carrow Road.
"We've been through every emotion going. A sending off, extra-time, penalty kicks - it's everything you want in a game of football," said Adams, who played more than 200 games for Norwich between 1994 and 1999.
"It was a fantastic turnout from the fans, I can't thank them enough. It's something the lads will never forget. We're delighted to get through."
Norwich's only ever FA Youth Cup final success came in 1983, when they beat Everton 6-5 on aggregate, with a side that featured Jeremy Goss, who would score one of City's most famous goals against Bayern Munich in the Uefa Cup in 1993, and Louie Donowa in midfield.
The two-legged final will be against either Liverpool or Chelsea, with the Blues 2-0 up ahead of Friday's return match.