The Pittodrie outfit defeated Inverness Caledonian Thistle on penalties at Celtic Park on Sunday to claim their first trophy since 1995.
Neither side could find a goal over 120 minutes, but Aberdeen held their nerve in the subsequent shootout to run out 4-2 winners after Billy McKay and Greg Tansey had failed from 12 yards for Inverness.
A powerhouse of Scottish football in the 1980s, Aberdeen were roared on by the majority of the 51,143 supporters in attendance.
Anderson and his team-mates have also displayed good form in the league this season - they sit second behind runaway leaders Celtic - and are into the semi-finals of the Scottish Cup.
Assessing the importance of Aberdeen's victory over Inverness, the defender told BBC Scotland: "It is the first step in what we hope will be more successful times for Aberdeen.
"It has been a long time waiting for it, so we just need to kick on now.
"I think everybody was expecting 25-30,000 fans to come here with us, but to get 40-odd thousand is incredible and definitely helped us."
Aberdeen manager Derek McInnes, who penned a contract extension earlier in the week, felt his side deserved to prevail.
"No matter how we've won it, we've won it and deserved to win it," he added.
"I thought we got the best of the first half in terms of controlling it and opportunities, but I thought the second half and extra time was very even.
"We have won a tight, fiercely contested match and we've shown enough bottle and composure at the penalty kicks.
"Everybody's doubted whether we can do this, but we've handled it every time there's been a question mark hanging over us.
"These supporters deserve some success."
