The 23-year-old became the first Reds player since Ray Kennedy in 1974 to score in his first three matches for the club with a close-range effort in the 5-0 demolition of Norwich.
Sturridge has already started to provide much-needed support to Luis Suarez, who took his tally for the campaign to 20 against the Canaries, and has shown he can link up with the Uruguay international.
In addition to that, Carragher believes the former Chelsea and Manchester City forward has given everyone a reminder they have to maintain top form at all times.
"He's done very well. He gives us a bit more pace and strength up there," said the veteran centre-back.
"He has a presence about him, he looks a really good player and has linked up well with Luis playing the slightly deeper role.
"He has got three in three games so he's made a great start and he's lived up to the expectation so far.
"But it is not just about goals - that's an important thing for a striker - but he's got great movement and pace.
"When he came in it gives everyone a lift and raises standards all around.
"The other strikers have to up their game to try to displace him and get in the team.
"It is early days for him but he has started well and it looks like a good decision to bring him in."
The blossoming partnership with Suarez, who occupied a deeper role behind Sturridge, has shown signs it can be profitable.
And Carragher knows it can only benefit a team who have been reliant on the South American livewire for so long.
"If you look at Luis' goalscoring record it is one of the top in the league so if he is your main striker and he is one of the top scorers he is doing his job," he added.
"It then needs other people to chip in from deeper areas but with Luis playing in a deeper area you still expect him to get goals and with Daniel up top he is going to get goals as well.
"No matter where Luis plays he is a world-class player but it was nice to have an extra body (in Sturridge) up there."