The Venezuelan defender, on loan from Brentford's west London rivals Fulham, lashed home in stoppage time to give Boro a precious advantage.
Andre Gray had earlier capitalised on a blunder by Boro goalkeeper Dimi Konstantopoulos to cancel out Jelle Vossen's first-half header.
The clash was outgoing Bees boss Mark Warburton's final game at Griffin Park, and his dream of signing off at Wembley is now hanging by a thread.
Boro chief Aitor Karanka took his players to Marbella this week to prepare for the clash.hey still looked in holiday mode in the opening stages, though, allowing Bees dangerman Alan Judge room to whip in a fierce shot which Konstantopoulos tipped onto a post at full stretch.
But the visitors, watched from the stands by Jose Mourinho - to whom Karanka was assistant at Real Madrid - took the lead after 25 minutes.
Vossen slipped unnoticed between two Brentford defenders to meet Adam Clayton's cross and guided his header across David Button and into the corner of the net.
Bees keeper Button then kept his side in the tie with two fine saves in the space of a minute before half-time.
First he got down well to turn Lee Tomlin's low drive wide at the near post, and from the corner somehow clawed out Vossen's close-range effort a hair's breadth before the whole ball crossed the goalline.
Boro's physical approach earned them four bookings in the first half and following one foul on Judge, Brentford came close to an equaliser but James Tarkowski nodded Alex Pritchard's free-kick narrowly over.
Gray missed a golden opportunity to equalise shortly after the interval when he blazed over after Jota had flicked on Judge's cross.
But the striker made amends in the 54th minute when he chased what looked like a lost cause from Jake Bidwell's hopeful ball into the channel.
Konstantopoulos came to meet the ball on the corner of the area but Gray charged down the keeper's clearance before spinning around and slotting the ball into an empty net.
Button was called into action again on the hour mark with another fine save, diving to his left to palm Albert Adomah's piledriver over the crossbar.
Konstantopoulos went some way to redeeming himself moments later when he pushed Jonathan Douglas' looping header over,
And Amorebieta stunned Griffin Park in the third minute of injury time George Friend headed down Grant Leadbitter's corner and the full-back's shot clipped Bidwell and flew into the net.