Man City will face the winner of the Chelsea-Leeds United semifinal next month.
Ismaila Sarr put Crystal Palace ahead in the 41st minute.
The Reds now hope for redemption as they play newly promoted side Burnley on Saturday.
Erik was appointed in April 2022 and led the club to two domestic trophies.
Meanwhile, Chelsea are also into the fourth round after the Blues pipped Brighton one-nil at Stamford Bridge.
Manchester City defeated defending champions Liverpool 3-2 in the fifth-round League Cup tie at the Etihad on Thursday night.
Manchester City will face Liverpool in the fourth round of this year’s Carabao Cup in a match featuring two of the remaining three big six clubs.
Hasenhüttl was appointed in December 2018, and played a key role in the development of the club infrastructure, identity and playing squad.
Mane, 30, joined Liverpool in 2016 from Southampton, winning the Premier League, Champions League, Club World Cup, Super Cup, Carabao Cup and FA Cup.
Kepa Arrizabalaga, who came in in the extra time for the penalty shoot out, missed Chelsea’s eleventh penalty, making the score 11-10 in favour of Liverpool.
