Mile Jedinak netted the only goal, heading home Jack Grealish’s corner after just 15 minutes at the Riverside Stadium.
Robert Snodgrass went close to doubling Villa’s advantage when his shot was tipped onto the post by home keeper Darren Randolph.
Boro struggled to create chances with substitute Fabio coming closest to an equaliser with a header saved by Villa keeper Sam Johnstone.
The two teams meet again at Villa Park on Tuesday night, where Steve Bruce’s hosts will be big favourites to progress after what was close to the perfect away day performance.
Just as they did back in December, Villa triumphed on Teeside thanks to a single goal.
Where Robert Snodgrass was the match-winner then, this time it was Jedinak who gave them an early lead they never relinquished.
Boro had plenty of possession but struggled to find a way through a resolute visiting defence, with Alan Hutton doing a sterling man-marking job on home talisman Adama Traore.
Villa also carved out chances. Indeed, had it not been for the brilliance of Randolph in keeping out Snodgrass’s first-half effort, their margin of victory would have been even greater.
The only worry for Bruce was an injury sustained late on by Ahmed Elmohamady, making the full-back a doubt for Tuesday’s return leg.
Bruce was able to field something akin to his strongest line-up, with Sam Johnstone back in goal just 24 hours after becoming a father for the first time.
The only real poser for the boss was who to field in central midfield alongside Conor Hourihame.
Bruce eventually went for Jedinak ahead of Glenn Whelan, due to the Australian international’s aerial prowess at set-pieces.
Just how strong the 33-year-old is in the air quickly became clear as he headed Villa into an early lead.
It was a devastatingly simple goal and one so surprising considering Boro’s reputation from dead ball situations.
Yet with ease, Jedinak was allowed to run unmarked to the near post to glance home Grealish’s delivery from Villa’s first corner of the game.
The goal rattled Boro, who had enjoyed the marginally better start. As the hosts struggled to find their feet, Snodgrass might have doubled Villa’s advantage but headed Adomah’s deep cross over the bar.
Eventually the chances began to come for the home side, the best ones to Britt Assombalonga.
But the striker shot over the bar and then wide after twice being set-up by Jonny Howson in quick succession.
He then brought the first save of the game out of Sam Johnstone with a header, admittedly straight at the Villa keeper, from Adama’s curling cross.
Johnstone’s biggest worry of the half came courtesy of a misdirected Hutton header which almost sneaked inside his own far post.
Instead it was opposite number Randolph who pulled off the heroics to keep his team in the tie.
Snodgrass looked to have increased Villa’s advantage with a trademark, curling effort but the keeper somehow touched it onto the post.
The visitors were content to sit back at the start of the second half and Boro struggled to create chances.
Assombalonga dragged an effort well wide of goal from outside the box, though Johnstone was fortunate to escape when he spilled a Howson cross, with Ryan Shotton unable to capitalise.
Boro were forced into a change just past the hour mark when Daniel Ayala was forced off.
His replacement, Fabio, almost had an instant impact but having got in behind his pull-back eluded everyone in the box.
Only in the final ten minutes did the hosts begin to pile on the pressure.
Johnstone was alert to save a Fabio header which threatened to creep inside the near post, while Mo Besic also slammed a shot wide.
In stoppage time, substitute Birkir Bjarnason dived in to deflect a Fabio shot wide as Villa held on.
15 - GOAL Mile Jedinak gives Villa the perfect start, heading home Jack Grealish’s corner.
42 - OFF THE POST Almost a second for Villa but Darren Randolph brilliantly tips Robert Snodgrass’s curling shot onto the post
Boro (4-1-4-1): Randolph, Shotton, Ayala (Fabio 64), Gibson ©, Friend, Clayton, Adama, Howson, Besic, Downing, Assombalonga (Bamford 76) Subs not used: Leadbitter, Cranie, Fry, Harrison, Dimi (gk).
Villa (4-1-4-1): Johnstone, Elmohamady (Whelan 88), Chester, Terry, Hutton, Jedinak, Snodgrass, Grealish, Hourihane, Adomah (Kodjia 71), Grabban (Bjarnason 80) Subs not used: Samba, Hogan, Onomah, Bunn (gk).