Boro had a few half-chances to level things up but were unable to take them, meaning just a third defeat in 15 outings to leave their promotion hopes looking bleak ahead of the return match at Villa Park on Tuesday.
The decisive moment of the first leg arrived early on as Jedinak flicked a front-post header into the far corner after being given time and space to meet Jack Grealish's inswinger.
The Villans took control of the match for the next 15 minutes, coming close to a second through attempts from Grealish and Jedinak, the latter having another header saved by Randolph.
The striker twice got between two defenders, only for his left-footed finishes to end off target, before directing a close-range header right at Sam Johnstone in the opposition goal.
Villa managed to hold firm and it was they who would have gone into the break with a two-goal lead if not for Randolph, who produced a stunning fingertip save to help Robert Snodgrass's curler onto the frame of the goal.
Boro dominated possession for large parts of the second half without truly troubling the visitors, waiting 20 minutes before creating their next opening.
When the Smoggies did get a rare shot on target, Ben Gibson did not know a great deal about it as the ball bounced off him and trickled towards the goal-line, where Johnstone was waiting to collect.
No team in the English Football League have conceded fewer goals than Villa in the final 15 minutes of games this term, but it was their net that looked more at threat of being breached late on.
Ryan Shotton headed a corner back across goal for Fabio to force Johnstone into a routine save, and the Brazilian full-back had a shot blocked behind soon after.
In the end, though, Villa were able to see things through to match their 1-0 win here during the regular campaign, giving them a big advantage heading into next week's second leg.
MIDDLESBROUGH (4-2-3-1): Randolph; Shotton, Ayala (Fabio 65'), Gibson, Friend; Clayton, Besic; Traore, Howson, Downing; Assombalonga (Bamford 76')
ASTON VILLA (4-2-3-1): Johnstone; Elmohamady (Whelan 88'), Chester, Terry, Hutton; Hourihane, Jedinak; Snodgrass, Grealish, Adomah (Kodjia 71'); Grabban (Bjarnason 81')