After a 0-0 draw at Anfield, Arteta’s side won the penalty shootout 5-4, with Maitland-Niles amongst the scorers from 12 yards.
Mohamed Elneny missed Arsenal’s third penalty, which momentarily handed the Reds the advantage.
However, Divock Origi missed the home side’s next spot-kick after a save from Bernd Leno, and Harry Wilson missed Liverpool’s sixth – also falling victim to the German’s heroics.
In the fourth round of penalties, Maitland-Niles stepped up, very casually and slowly trotted up to the ball, before very calmly slotting it down the middle, past Adrian.
As the 23-year-old was running back to his team-mates on the halfway line, the Gunners defender and the Reds’ goalkeeper exchanged a smile – with some words clearly said beforehand.
Arteta, speaking in his post-match press conference, joked about Maitland-Niles’ penalty run-up almost giving him a heart attack.
The Arsenal boss said: “I told him in the dressing room, I’m going to have a heart attack with his body language, Jesus Christ!”
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Despite the casual run-up, there was nothing relaxed about the celebrations from the north London side – who clearly feel that this is another great opportunity for more silverware this season.
However, they will face a tough task in the quarter-finals, as they welcome Manchester City to the Emirates Stadium.
Arteta will face his old boss, Pep Guardiola, who he spent three years working under at the Etihad Stadium, prior to his appointment as Gunners manager.