Sturridge put Liverpool ahead at Villa Park, before James Milner’s free-kick crept in.
Goals from Emre Can, Divock Origi and Nathaniel Clyne made it 5-0, before Kolo Toure headed in a sixth to add further gloss to the scoreline and heap misery on Villa.
The result left Liverpool eighth in the Premier League table and nine points behind Manchester City in fourth.
However, Carragher believes that Liverpool are still lacking in quality, despite their convincing win over the Premier League’s basement club.
Speaking on Sky Sports after the game, former Reds defender Carragher said: “Brendan Rodgers could look back and say the injuries to Daniel Sturridge maybe cost him his job. That’s how much he elevates this Liverpool team.
“Without him and Coutinho, it’s an average team and we’ve been watching that for the last few months.
“We’re expecting Jurgen Klopp to come in and make a difference, but I think he said in the week he’s not David Copperfield. You need top players.
“That’s what Liverpool have. They haven’t got that many of them, but the one true one they have is too injury-prone and that’s the problem for them.
“If they can keep him fit between now and the end of the season, you would expect a good run.”
Liverpool, who were dumped out of the FA Cup by West Ham United in midweek, take on Manchester City in the League Cup final at Wembley later this month.
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