A young Liverpool side took an early lead through Shkodran Mustafi's own-goal before the visitors scored three times in 17 minutes to put themselves in pole position.
James Milner's penalty before half-time put the Reds back in contention before Ainsley Maitland-Niles made it 4-2 not long after the interval.
An Alex Oxlade-Chamberlain screamer and a Divock Origi strike from the outside of the box sensationally put the Reds back on level terms. However, Joe Willock put Unai Emery's side 5-4 up with 20 minutes to go.
With seconds remaining, youngster Neco Williams' cross found Origi, who turned the ball home with an excellent acrobatic finish.
The Reds were flawless with their penalties as Caoimhin Kelleher made a crucial save, sealing victory and progression to the quarter-finals.
They could get another opportunity to make a mark in the last-eight, with Liverpool drawn against Aston Villa away from home.
With Liverpool in action twice a week, every week, until January, there remains few alternative dates that would work for the Reds.
Speaking after his side's win over the Gunners, Klopp admitted: “If they don't find a place for us, an appropriate place, not 3am on Christmas Day, then we don't play.
“But you have to think about these things, if you have a fixture list where one team cannot be part of all the games, then you have to think about the fixture list.
“And hopefully it starts at one point, hopefully, it starts now. I really think it's fair that this problem is obvious now.
“But we will not be the victims of this problem, we played tonight, we wanted to win it, we did that.
“If they don't find a proper date for us, then we cannot play the next round and whoever is our opponent will go through, or Arsenal plays—I cannot change that.”
Further details are expected in the coming weeks, with Liverpool's participation looking increasingly unlikely.