Mile Jedinak's header in last weekend's first-leg against Boro decided the tie, after the teams played out a tense 0-0 draw in Tuesday's second leg at Villa Park.
The playoff final will be held at Wembley Stadium on May 26 where Villa will face Fulham who overcame Derby County 2-1 on aggregate on Monday.
"We just shaded it over the two games, with a goal from a set-piece", Bruce said. "It's hard for me to say that but that's being brutally honest". It means so much.
Bruce has led two teams through the playoffs to the Premier League but expected a tight encounter against the Londoners.More news: TN to sue Purdue Pharma for role in opioid crisis
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"It was intense. I think our game plan worked very well so the manager deserves a lot credit for that".
"Great team spirit amongst the boys and like I said before, one more step and it will be the dream".
Randolph then came to the fore as he made two fine saves from Lewis Grabban before getting fingertips to a long-range effort from Jack Grealish.
Sam Johnstone could have been shown a red card by Mike Dean in the final moments of the game when he handled the ball outside the penalty box but the referee decided John Terry could have been able to get around and gave the goalkeeper a yellow instead.