The holders progressed thanks to second-half goals from Shaun Rooney and Ali Crawford.
Davidson was delighted for the fans who had missed out last season as the club lifted the League Cup and Scottish Cup.
He said: "It's one of those we said we'd do for the supporters. I thought they were brilliant for us. They kept us going all the way through.
"I thought Dundee, in the first half, probably had the better chances.
"In the second half we weathered the storm but once we scored, I thought we settled down, soaked up pressure and then hit them on the counter.
"It's just great to go back to Hampden really. It's fantastic. It's a great achievement by a club like St Johnstone to get there again.
"It is really important to have the fans there as last year they were not allowed to come."
Dundee boss James McPake is fed up sounding like a broken record after his side failed to score for the fourth game running.
He said: "It was a hard one to take, particularly after the chances we had in the first half.
"Again, it is a broken record and I am coming out and saying the same things.
"When you are creating gilt-edged chances, you have to take them no matter what level you are playing at but particularly against better sides.
"You know that when they get a chance they are going to hurt you.
"That's what happened. The first one was a poor goal. It was a set-play and it was ridiculous we lost that goal.
"It was really poor defending and the same at the second goal."