The former St Johnstone and Bristol City boss took charge at Pittodrie last year and the Dons have now claimed their first trophy since winning this event back in 1995.
Cheered on by 40,000 fans at Celtic Park, Aberdeen eventually won 4-2 on penalties after neither side could score in 120 minutes of football, although Dons skipper Russell Anderson did hit the post.
And McInnes said: "I'm absolutely ecstatic. You can see what it means to so many people.
"We were well aware when we took the job of the need and desperation to get a trophy again and to get Aberdeen successful.
"We have got a brilliant board and a great chairman, and there are a lot of staff who have had to endure Aberdeen for a long time, working hard for little success.
"But the supporters came here in huge numbers because they believe in the team and I saw how hard my players worked for this.
"It was a tight final. I thought we had the greater opportunities and, once it comes down to penalty kicks, you've got to back yourselves and take them with confidence. I thought our penalties were different class and thankfully Jamie (goalkeeper Langfield) has also managed to save one.
"It's unfortunate for Inverness, who had a good final. It was tight, as I said, but the overriding thing for us was just to win the game, no matter how we did it."
McInnes, who in midweek signed a new deal at the Pittodrie club which takes him to 2017, revealed an unconventional attitude to the penalty shoot-out.
"I said to the players at the penalty kicks that it is not a lottery," he said.
"I recalled the game against Alloa in the second round when the standard of penalty kicks was terrific and we didn't miss one.
"We practised them all week and I told them to go and be confident."