The Dons lifted a trophy for the first time since 1995 after they prevailed 4-2 in the shoot-out after the match had finished goalless following extra time.
Anderson, who is in his second stint with his home-town club, said: "It's an incredible feeling. It's been a long time coming.
"It was a very nervous, tense day. The longer the game went on you just wondered if it would go to penalties. The players have taken some great penalties, and I think we deserve it."
Aberdeen could also have another cup final to look forward to as they are in the semi-finals of the Scottish Cup, and Anderson hopes this result can help them go on and clinch a trophy double.
"You'd like to think so, it's not going to hinder us anyway. Now we've won one it will give us confidence to finish the season strongly," added the defender.
Midfielder Barry Robson was equally thrilled with the win and was not bothered that they needed a penalty shoot-out to lift the title.
"We've won the game, that's the most important thing, I don't care if it's penalties, I don't care how we play, we've just won the cup," the 35-year-old said.
"It's great, it's brilliant. It's been a long time coming for this club.
"It's sensational, you can see what it means to the people."
Striker Adam Rooney, who scored the decisive penalty, said: "I tried not to think of the fans watching, I was just trying to get a good connection with the ball and thankfully it went in.
"It was nice to grind out the win in the end and reward the fans who came here.
"I think we could have sold 50,000 tickets, as there seemed to be so many in the city who couldn't get them, which is incredible support and they did give us that lift.
"We have to make sure we build on this now."