Confusion in the Gers defence about a possible offside at Hampden Park on Sunday allowed Dons striker Bojan Miovski to convert a Matty Kennedy cross and the Ibrox men had to use the interval to plot a comeback.
Former Dons midfielder Ryan Jack drove in a deflected leveller in the 61st minute before Pittodrie skipper Anthony Stewart was shown a straight red card by referee Nick Walsh for a shocking tackle on Gers attacker Fashion Sakala when no danger was imminent.
Substitute Kemar Roofe put Rangers ahead from close range four minutes into extra-time and despite a nervy ending against the 10-man Pittodrie outfit, Rangers held out to set up an Old Firm clash in the final on February 26.
Midfielder Tillman, on loan from Bayern Munich, said: "The manager said at half-time that it is a mentality thing. Even in extra-time it is always a mentality thing.
"He also said that we played very well and we just had to come back and score one goal and then the second.
"You always have to work hard and keep in your head to stay calm and do what you can do.
"I think we dominated the whole game but conceded an unlucky goal, or we didn't defend well, but we came back again."
The 20-year-old USA international looks forward to facing Celtic at the national stadium with the first major trophy of the season up for grabs, although he described the playing surface at Hampden Park as "not that great."
He said: "Everyone wants to be there. As a team, we want to win titles and this is a big step towards one of them.
"Obviously we are nine points behind in the league and this is the first title we can compete for so we will try to win that."