Jurgen Klopp's side, finalists in the competition last season, slumped to a disappointing 2-0 Premier League loss at Burnley on Saturday but an upset was never on the cards at the Pirelli Stadium.
Origi put Liverpool ahead after just 14 minutes, tapping in from close range after Sadio Mane's trickery created a simple finish for the striker.
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, on as a 64th-minute substitute for Firmino, compounded the hosts' misery with two goals in the space of five minutes in the closing stages - his first strikes of the season - with injuries to Emre Can and Origi the only sour notes for the visitors.
Liverpool had progressed in all of their previous 12 League Cup ties against lower-ranked opposition and there was little chance of that record ending in their first competitive meeting with Burton.
Origi made his first start of the season as part of a strong Liverpool team and he forced Stephen Bywater into a smart save at the near post after just five minutes.
Bywater was soon in action again as the visitors opened Burton up but Firmino's header was pushed around the post.
The early pressure paid dividends in the 14th minute, though.
But Liverpool were gifted a second when Bywater's poor throw went straight to Mane and he screwed the ball into the path of Clyne, who in turn produced an excellent cross on the run that Firmino nodded home.
Burton enjoyed a good spell of possession after half-time but Nigel Clough's men struggled for penetration and Liverpool's third goal arrived on the hour-mark when Milner's whipped corner flicked off Can and bounced in off Naylor to extend the lead.
Sturridge added a fourth goal with 13 minutes to play, the substitute tapping in at the far post from Milner's low cross after a one-two with Mane.
And Sturridge was soon on target again. Mane did well down the right again and when he picked out Sturridge in the box, the striker made no mistake with a powerful left-footed finish to round off the scoring.
Liverpool's comfortable win was marred by both Can and Origi picking up injuries, however, with the closing minutes seen out with 10 men.