The 24-year-old, who helped Argentina lift the trophy in Qatar, scored his first goals since returning to action for the Seagulls on Saturday.
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On as a second-half substitute, Mac Allister bagged a brace to send Brighton into the fourth round.
His first strike came at a crucial time for the visitors as he restored Brighton's two-goal lead and made the scoreline 3-1 in the 58th minute.
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And the ex-Boca Juniors loanee found the bottom right corner in style with a stunning backheel flick.
Pervis Estupinan fired a low shot at goal from the edge of the box, and Mac Allister instinctively guided the ball to the far corner to leave Boro goalkeeper Zack Steffen no chance.
He was quickly mobbed by his teammates who were left in awe by the audacity of his effort.
And Mac Allister wasn't done there, as he grabbed his second of the afternoon 22 minutes later.
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Found by Pascal Gross inside the box, Mac Allister swept the ball home into the bottom left corner.
He could have had a hat-trick in Yorkshire, but his free-kick from 25 yards out cannoned off the post.
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Brighton weren't done there, as they put the game to bed in the 88th minute through Deniz Undav as they ran out easy 5-1 winners.
And they did it without star man Leandro Trossard, who was left out of the squad after boss Roberto Di Zerbi stated he 'wants more' from the Belgian forward.
Speaking after the match to BBC Radio 5 Live, Mac Allister said: "[I'm] Living the dream, very happy.
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"But credit to the team today they played really well and we deserved the win.
"To be honest I wanted to come back, the club gave me to the fifth but I came back earlier because I wanted to train, play and share my medal with the club.
"I feel very happy to be a part of this club and to be in this city so I will continue to give it my all."
Commenting on Mac Allister's first goal, Di Zerbi stated: "He has the quality to score.
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"Before the World Cup I used him in the midfielder position - more responsibility to build the play.
"Now, he plays more higher, and I think playing higher he can help us to find the last pass to find the goals because he knows that position very well."
Asked about whether he has returned more confident, Di Zerbi remarked: "He is very strong in the head, and he didn't need to win the World Cup to have more confidence."