The Reds looked far from comfortable in their 1-0 win against Hearts, but the defender, who also labelled Raheem Sterling as one to watch, thinks they will improve soon
The Premier League outfit found it hard against Hearts in their Europa League play-off tie on Thursday evening, with John McGlynn's men edging the majority of the first half. However, an Andy Webster own goal gave the visitors a vital first-leg victory 12 minutes from time.
The win saw the Merseysiders successfully bounce back after a 3-0 loss to West Brom on the first day of the new league season - their worst opening result for 75 years.
Carragher admitted that the Baggies defeat was a setback, but stressed that the players are looking to make the transition to a slick passing side under the new boss as smooth as possible.
"It doesn't have to be a slow process," he told ITV Sport. "Pre-season's gone really well, obviously we've got a bit of a bump at West Brom.
"It was nice to come here, it was a difficult game, we've got the result now and now we look forward to Man City.
"It's a massive game, and as I said, a new manager coming in, finding his feet if you like and we're all getting used to each other.
"But it doesn't have to be slow, we want it to be as quick as possible."
The defender also applauded Hearts for their display, indicating the clash was just as tough as the Premier League's biggest teams, while he also cited the fantastic atmosphere in Scotland.
"It was a very difficult game and sometimes these games are more difficult than when you play so-called big names," he said.
"We knew they'd come out flying, it's a very tight pitch - there wasn't much space there for us to try and pass the ball - and that's what we tried to do.
"And at times early on we got caught with it, because they created pressure.
"But the atmosphere was fantastic and great credit to Hearts."
Raheem Sterling was a bright spark for Liverpool, particularly in the first half where he caused the Hearts back line continual problems with his pace. The 17-year-old winger also presented Fabio Borini with a brilliant chance to take the lead just before the break, but the Italian struck the post with his shot.
Carragher insists the teenager will be a top player at Anfield in the years to come, also stating that late substitute Adam Morgan is another one to watch from the club's youth system.
He added: "This club's got a great tradition of bringing young players through and we brought another one in there, Adam Morgan came through the academy and Raheem's got a few starts now this season.
"I think the way the manager wants to play is with wide men so it suits him down to the ground.
"He's come here, he's done really well and I think he's 17, 18, I'm not sure, but it looks like he's got a big future at the club."