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Midway through the half Liverpool threatened again with Adam Lallana creating some space in the midfield before firing a strike wide of the target.
Lee Tomlin had Middlesbrough's best chance of the game with six minutes of the first half left to play, but his powerful strike was saved by the keeper as the hosts held on to take their lead into the break.
They almost grabbed a second as Adam Clayton fired in a strike from outside the box that was kept out by Mignolet.
Clayton went even closer for Boro soon after when he saw his effort from a tight angle come back off the post.
Neither side were able to find the net in normal time and the game headed towards an extra 30 minutes.
Liverpool dominated throughout extra time but had to wait until the 109th minute to make the breakthrough, as Mario Balotelli's blocked shot fell kindly to Suso, who then placed a low effort past the keeper.
The Reds had looked set to claim the points, but in the third minute of stoppage time Middlesbrough were awarded a penalty when Patrick Bamford charged into the box and was brought down by Kolo Toure.
Bamford stepped up to take the spot kick himself and he kept his composure to slot a low effort into the bottom corner to send the game to penalties.
Bamford and Raheem Sterling both missed from the spot as the shootout headed towards sudden death.
Following a run of penalties that saw both keepers step up to score, it was Albert Adomah who finally missed the target, with Liverpool heading through to the next round.