The Reds were second best to the Citizens for large spells, riding their luck before scoring a late goal through Philippe Coutinho to force the match into extra time.
Jurgen Klopp's charges had their moments to win their first trophy since 2012 in extra time, but eventually lost out on penalties as Willy Caballero saved three spot-kicks for City.
And afterwards Carragher stated that he believes Klopp has a lack of quality at his disposal which will also make it difficult for Liverpool to win the Europa League this season, with his former club set to face rivals Manchester United in the last 16.
"A lot was riding on today [Sunday] as it's been a poor season," Carragher told Sky Sports.
"Not just for the club, but for the manager, if he's to be successful then they've got to win trophies and this was his first shot at it.
"However, he's aware there's problems in this squad. There's not enough quality, they're not good enough.
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"Where they are in terms of the other teams and what they can win in terms of trophies, this is it. They're not good enough to win the Premier League or the Europa League. They have Manchester United next in that tournament so it will be like a Cup final."
Coutinho's strike in the 80th minute at the national stadium was Liverpool's first shot on target in the whole match against City, as the Merseysiders found it difficult to create chances on the day.
One player who struggled to perform was Daniel Sturridge but while Carragher agrees the marksman found it difficult to shine, he believes Liverpool cannot afford to sell him as there are few viable alternatives available.
"Sturridge didn’t perform well. Liverpool need to keep him though as there are very few quality strikers out there," he added.
"He is one of Liverpool's few real technical players.
"They can't get the likes of a [Karim] Benzema. Benteke cost £30million and he couldn't even get on the pitch."