They booked a promotion showdown with Fulham on May 26 after a 1-0 aggregate win over Middlesbrough.
Steve Bruce's side drew 0-0 with Boro at Villa Park - which held a minute's applause for former defender Jlloyd Samuel after he was killed in a car crash - on Tuesday to progress to the Wembley final.
They won 1-0 at the Riverside on Saturday and Terry, who joined on a free transfer after leaving Chelsea last summer, saluted their achievement.
He told Sky Sports: "It's unbelievable and incredible. The support from the lads, the dressing room is incredible. There is a great team spirit among the boys. One more step and it'll be the dream.
"It was quite tense, I think our game plan worked superbly so our manager deserves a lot of credit for that. The boys have been superb and we're off to Wembley - one more to go.
"We'll not to get too excited, there's still one to go. It's a huge week ahead for us."
Boss Steve Bruce, who brought Terry to the club, also hailed the senior players after admitting he inherited a club in a mess following relegation when he was appointed in October 2016.
He said: "Eighteen or 19 months ago this club was in a mess but the experienced players have done the job for us and we're delighted about that.
"As a one-off, it's fantastic if you go up through the play-offs, although I would love to have been second and sitting on a beach now."
Boro failed to register a shot on target but Stewart Downing hit the bar after Sam Johnstone handled outside the area with three minutes left.
But boss Tony Pulis believes the goalkeeper should have been dismissed by referee Mike Dean.
"I think it's a disappointing decision from one of our top officials," he said.
"It irks me a little bit but the teams who finished third and fourth are going to Wembley and that's justification of what they've done over the season. We've just fallen short."
Before the game, Villa paid tribute to former defender Samuel who was killed in a car crash aged 37 on Tuesday.
The home side wore black armbands and there was also a minute's applause before the game while fans chanted the defender's name ahead of kick-off.
During the applause, Villa's big screens showed Samuel's two league goals for the club - both against Charlton during the 2003-04 season.
Register for free to continue reading
Registration is a free and easy way to support our truly independent journalism
By registering, you will also enjoy limited access to Premium articles, exclusive newsletters, commenting, and virtual events with our leading journalists
Already have an account? sign in
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies