The 35-year-old defender is in his second spell at the Pittodrie club but it was as a Dons' apprentice that he watched Aberdeen beat Dundee 2-0 in the League Cup at Hampden in 1995.
Such was the waiting time between trophies that the 4-2 spot-kicks win over Caley following a goalless 120 minutes at Parkhead resulted in some euphoria among the Dons players and 40,000 and more of the Red Army who made the journey to Glasgow from the Granite City.
However, Anderson knows McInnes will soon have his squad preparing for the Scottish Premiership clash with Kilmarnock at Pittodrie on Saturday where the second-placed Dons will try to stay ahead of chasing Motherwell.
"It is an exciting time at the moment but we also know that if we don't try to keep progressing then we can pretty quickly fall flat on our face," he said.
"The season is not finished yet and I am sure the manager will be the first to remind us of it, that we have a game next Saturday.
"That is the way that he works, that when the game against Kilmarnock comes round, we will be ready for it."
Immediately after the on-pitch celebrations, McInnes, who, along with assistant Tony Docherty, signed a new deal last week which ties them to the Pittodrie club until 2017, targeted a cup double as the Dons look forward to the William Hill Scottish Cup semi-final against St Johnstone at Ibrox next month.
Former Sunderland stopper Anderson, who won 11 caps for Scotland, concurred with his gaffer's ambition but with typical caution.
He said: "We are in a good position now with the semi-final of the cup but we certainly won't be taking anything for granted because as soon as you do that you will become unstuck.
"But hopefully this cup win will give everybody at the club confidence to believe that we can go on and have more successful times.
"First and foremost, the squad has ability. Also, a desire to win that feeds from the manager. That's where it starts from, him and Tony.
"Every game we go into, they are desperate for us to win and have the mentality, that we are focused, going to compete and give a good account of ourselves.
"When you put those two things together, it shows what can be achieved."
Adam Rooney's reward for joining the Dons from Oldham in January came when he fired in the winning penalty and the former Caley striker, 25, is also looking forward to more success under McInnes.
"The club is going in the right direction," said the Irishman.
"It has been brilliant since I came here.
"We managed to end Celtic's unbeaten run, we managed to get to a cup final and win it and we are in another semi-final.
"So the club is definitely going in the right direction at the moment and we have to make sure we keep working hard to keep that up."
Meanwhile, Aberdeen say they will stage an open-top bus parade with their trophy through the city on Sunday. Final discussions are on going with Aberdeen City Council but the parade is expected to start at 1pm.