Mile Jedinak's fine glancing header from Jack Grealish's corner in the 15th minute put Steve Bruce's men in the box seat for a trip to Wembley.
Aston Villa 1-0 Middlesbrough ⚽️🔥— EveythingFootball (@EvfFootball) May 12, 2018
Full Jedinak Goal! pic.twitter.com/GyomsdQlhJ
Villa had been considered favourites to advance from the tie after finishing seven points above Middlesbrough in the table and they cleared the first hurdle with a clean sheet intact.
Captained by John Terry, who lined up alongside James Chester in the heart of defence, the value of the visitors' experience showed as they restricted the hosts to few real chances and just two shots on target.
Sam Johnstone was required to make a smart close-range save from Britt Assombalonga soon after the opener, before an outstretched Darren Randolph brilliantly turned a curling Robert Snodgrass effort against the post late in the first half.
That was the closest either side came to changing the scoreline as they now switch attention to the second leg at Villa Park on Tuesday.