After going behind to Bojan Miovski’s cushioned finish on 41 minutes, the Light Blues battled back with man of the match Ryan Jack equalising after the hour mark before super sub Kemar Roofe scored the winner in the fourth minute of injury time after terrific wing play from Scott Wright, who also went on from the bench.
Speaking post-match at the national stadium, Beale told RangersTV: "I'm delighted for the fans, I'm delighted too for the players, I'm delighted for everybody.
“Semi-finals are there to be won and we've had other days out here where we haven’t got what we deserved, but I thought today we did.
“It was a strong performance and we found ourselves 1-0 behind. That took a lot of character and mentality and I've just said to the boys in there: ‘Well done, you've come back from 1-0 down in the semi-final where you didn’t rush it, you kept playing our football and we scored two excellent goals’ - so I'm delighted.
“We maybe could have scored one or two more and I have to say fair play to Aberdeen, they put on a really strong performance today as well. It was a tough challenge but I think it was right that we won and I'm just delighted for everybody.
“We won the game and we now get to come back here in a month's time and reward our fans for their fantastic support today. It would have been wrong if we had sent them home having not won that game.
"At half-time we spoke about how we wanted the second forty-five to go. We wanted to keep playing and we felt we were the more dominant team in the game and that we had to improve out last pass and shot. If we went on and scored we felt we would run the legs down a little bit.
“Obviously the sending off takes them down to ten men but in the whole game the pitch didn’t help. We are playing against a low block on a bad pitch and it is a bad pitch to play on.
“It was tough on the legs but the mentality and character of this group is outstanding. It has been since I have been back but it has been from the months before that as well.
“I am absolutely delighted for everyone. As I said, semi-finals are there to be won and we won, we park it now and we go to Killie looking for three points on Wednesday."
The manager admits he was thrilled to see his substitutes Wright and Roofe go on and have such an important impact on the semi-final.
He added: "I keep saying to Scott Wright that he is a much better player when he is direct and he takes the one v one.
“We spoke about it in front of the group in the week and I want the other players to force him to do that more as well. And he likes it at Hampden, doesn't he? He scored here in the cup final and it was a fantastic run today and Kemar with the finish.
“That's what we are looking for and as we get everybody back and fit you will see the strength of our Rangers squad."
Roofe didn’t finish the game as he walked off holding his shoulder in the second half of extra time following a collision with an Aberdeen defender, and Beale provided an update, saying: "It is not a dislocation I am told. I am told it's a heavy landing so he'll have an x-ray on it tomorrow.
“It is very sore at the moment so he can't catch a break at the minute. He comes back, scores the winner in a big game and then has to go off injured. So I feel for him and I feel for Antonio Colak who has barely played 45 minutes under me as well.
“If we can get those two fit and around the squad with the way the other attackers are playing, it gives us real optimism for the future."