Villa booked a Wembley showdown with Fulham on Saturday, May 26, after preserving their one-goal advantage following a goalless draw at Villa Park.
Steve Bruce's side claimed a 0-0 draw at Villa Park on Tuesday, despite a nervy finale, to progress following their 1-0 victory at the Riverside on Saturday.
They will face Fulham - who beat Derby - in the final at Wembley on May 26 as they look to end a two-year Premier League exile.
Boro spent around £50million in search of an instant return to the Premier League but sacked Garry Monk after just over six months in charge.
Replacement Tony Pulis took them to fifth but they now face another season in the second tier.
In a game short on clear chances, Conor Hourihane headed off the line from Boro's Ryan Shotton while at the other end, Darren Randolph saved from Lewis Grabban.
Stewart Downing's free-kick hit the bar late on for Boro - after Villa goalkeeper Sam Johnstone was lucky to escape a red card for handball outside the area.
But Mile Jedinak's first-leg header settled a tie which Villa rarely looked like losing on Tuesday.
Their early energy was contrasted by Boro's nerves and the visitors survived when Hourihane charged down Randolph's clearance.
James Bree had already shot wide during Villa's bright start as they tried to kill the tie off.
Slowly, though, Boro gained momentum without opening Villa up in a scrappy first half-hour.
Despite losing their early punch Villa still created openings and George Friend produced a splendid tackle to thwart the onrushing Hourihane.
Sky Bet Championship play-off final winner (odds via Sky Bet)
- Fulham - 4/6
- Aston Villa - 11/10
And seven minutes before the break Grabban nodded down for James Chester to shoot narrowly wide.
But Boro nearly levelled the tie on 40 minutes when Shotton's header was cleared off the line by Hourihane.
Nerves began to grip Villa Park and Boro's Mo Besic dragged a shot wide seven minutes after the break as the game opened up and the teams raised the tempo.
Grabban was denied at close range by Randolph on the hour and Villa's confidence grew.
There was a growing sense Middlesbrough had run out of ideas and Britt Assombalonga was replaced by Patrick Bamford.
It was Villa's tie to lose and Grabban nearly sealed it when Randolph turned over his 30-yard effort with 20 minutes left.
The Boro goalkeeper then palmed away Jack Grealish's curling strike.
Yet Villa survived a scare with three minutes left when Johnstone was only booked for handling Adama Traore's shot outside the area and Downing hit the bar from the resulting free-kick.
Villa captain John Terry hailed his side's displays over the course of the 180 minutes but is already focusing on the Fulham clash.
Terry told Sky Sports: "Two very good performances, away and at home tonight. The boys are different class. It was important in the first half we didn't concede, another clean sheet in two games is so very good.
"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 can't get too excited, there's still one to go. Playing two games in the space of four days, it's important to get a couple of days' rest - which the manager's hopefully going to give us - and then back on it. Huge week ahead for us."
Influential midfielder Jack Grealish added: "We fought well tonight, it was always going to be a hard game coming back here with a 1-0 lead but we've done well and we're looking forward to the final.
"(The game against Fulham) is going to be a hard game. We've played them twice this season; we've won one and lost one, so it's all to play for at Wembley. It's a one-off game but I look at our team and we've got enough quality in there to go out and beat them."
Boro boss Tony Pulis rued his side's inability to create opportunities when they were on top.
He told Sky Sports: "I think we've had good spells in both games and in those spells we have to create a little bit more and score goals. That's been the difference really: our set plays have really cost us over two games.
"We've got to dust ourselves down, thank all our supporters for the way they've supported the team and we've got improve a little bit if we can next season."
Pulis admitted Fulham and Villa deserve to contest the final, saying: "The teams that have finished third and fourth over a season have got to Wembley.
"I can accept over a long and hard season that they've been the third and fourth best teams in the league, so you have to take it on the chin. It's difficult for me to say that but that's being brutally honest."
On the decision to only book Villa goalkeeper Johnstone, Pulis added: "I was really disappointed. Everybody can see it and everybody understands the rules. You need the referee to make the correct decision and I don't think he made the correct decision."