After a dour first half at Hampden Park, the Easter Road side were denied an opener in the 49th minute when VAR ruled attacker Martin Boyle was offside before he had the ball in the Dons net.
It looked like Hibs had gained an advantage in the 75th minute when Aberdeen defender Jack MacKenzie was sent off by referee John Beaton for picking up a second yellow card for a silly push on Hibees defender Lewis Miller.
However, it was Aberdeen who responded positively when striker Bojan Miovski broke clear three minutes later to fire in the winner to set up a December 17 final meeting with either Rangers or Hearts, who play in Sunday’s other semi-final at the national stadium.
Boss Robson said: “First and foremost, I’m so proud of my players.
“We’ve had to play four games in nine days and we’ve had a day’s less rest than Hibs. They haven’t had the games we’ve had.
“So to fight like that, it shows the real spirit and determination that the group has got.
“We’ve been on our travels a lot and had a really hard game at Killie and then we were away again to Motherwell.
“Now we’ve come to Hampden so for me, to get that character and spirit is everything to me and more important than talent or ability.
“We couldn’t play the way we wanted with four games in nine days. With all the games we’ve had, it’s been difficult.
“We had to be tactically clever today – we just couldn’t go after them the way we wanted to. It’s impossible.
“But we were tactically good and the players performed.
“There wasn’t a lot in the first half. They started the second half well and we had a man sent off.
“Listen, Bojan could have scored three goals so for me, it’s a semi-final and it’s about winning.
“And coming on the back of all that travel, what a performance that is from my players.
“What a spirit, what a group. They are fighting for their club and they’re in a final.”
On MacKenzie’s second yellow, Robson said: “I never saw Jack’s second booking, I was looking at the other side. I’ll need to have a look back at it.”
Hibs boss Nick Montgomery was “definitely hurting” and claimed two decisions went against his side – the offside call on Boyle and the decision by VAR not to intervene when Dylan Vente went tumbling in the box under pressure from Aberdeen goalkeeper Kelle Roos, with referee John Beaton taking no action.
He said: “I thought it was a game we controlled for large parts and there were a couple of incidents that didn’t go for us.
“All credit to Aberdeen, it’s one counter-attack and a moment in the game where we have to be smarter. We have to slow the game down. It’s possibly a foul on Lewis Miller.
“I’m pleased with the performance. I felt like the decisions didn’t really go our way.
“At Martin Boyle’s goal, if it takes five or six minutes to deliberate whether he is offside then it is not clear and obvious in my opinion.
“I have just looked back at it and I don’t see how he is offside. The lines are virtually touching the defender and him. It doesn’t go in favour of the attacker.
“And then the penalty incident. How that wasn’t looked at on the screen. From where I am the keeper has spilled the ball and Dylan Vente got there before him and touches the ball and is virtually rugby tackled in the box.
“For that just to be waved away and not even looked at… the rules of the game are you can’t make contact with an opposing player in the box.
“I felt at least one of those two incidents could have gone our way and that would have changed the reflection of the game.
“I thought we played well enough to get to the final. So very disappointed.”