With Liverpool’s only available defensive midfielder, Emre Can hurting his ankle, the Reds victory still ended with concern.
Jordon Henderson played out of position in the loss against Burnley on Saturday, and with Lucas Leiva injured, Liverpool has problems.
Nevertheless, they are through to the next round of the League Cup against easy opposition on the night.
The impressive Sadio Mane teed up Divock Origi for the Reds’ opener on 15 minutes and Roberto Firmino nodded home their second (22).
Mane and Nathaniel Clyne then combined well for Firmino to double the lead before the break and an own goal from Tom Naylor in the second half ended any hope of an unlikely comeback from the Championship side.
Sturridge got himself on the scoresheet on 78 minutes, following a lovely sweeping move that saw Milner drifting from left to right, combining with Mane before laying on a simple, low cross for the No. 15 to touch home.
He had his second soon after, with Mane driving through the Burton defence on the right to play a crisp pass into Sturridge’s path, with the striker making no mistake in the box.
Klopp took Liverpool to the final last season, losing to Manchester City on penalties, after succeeding Brendan Rodgers two months into the campaign and his team are on the Wembley trail again.