Adrian made the only save in a 5-4 penalty shoot-out win for Jurgen Klopp's men in Istanbul, where the Premier League giants drew 2-2 on Thursday (AEST).
The shot-stopper only signed for Liverpool on August 5 and he delivered a huge moment on his starting debut, playing due to star goalkeeper Alisson's injury.
Amid rumours Adrian would be looking to move to Merseyside, Carragher – who won two UEFA Super Cups with Liverpool – offered his house for very little.
"Adrian can have the house for f*** all!!," Carragher tweeted.
Sadio Mane's brace had put Liverpool ahead in extra time after Olivier Giroud's opener, but Jorginho's 101st-minute penalty forced a shoot-out before Adrian's heroics.