Manchester City took the lead in the 49th minute of the final when Fernandinho fired past Simon Mignolet at the near post.
Liverpool battled hard, however, and drew level seven minutes from full time when Philippe Coutinho fired home to force extra-time.
The game remained goalless in extra-time as both sides had chances at Wembley before the penalty shootout.
And Willy Caballero saved three penalties – from Lucas Leiva, Coutinho and Adam Lallana – as City went on to lift their first trophy of the season, triumphing 3-1 in the shootout, with Yaya Toure netting the winner.
Sturridge did not take a spot-kick, and Carragher admitted that he was baffled by the decision after the game.
Speaking on Sky Sports after the final whistle, former defender Carragher said: “He had cramp towards the end, but unless he’s got an injury, I don’t know why he’s not taking a penalty.
“I don’t know why someone like Lucas is taking the second penalty. You want your best penalty takers [first], you think they’d be more chance of Daniel Sturridge scoring than Lucas Leiva.
“The fact that a penalty shootout has gone on and Daniel Sturridge hasn’t taken one for Liverpool is stupid.”
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