Fernando Torres extended Liverpool's recent revival with two goals that helped Rafael Benitez's side move up to third place on the back of a 3-2 victory at West Ham United.
Torres's excellent finishes, sandwiched between a close-range effort from Dirk Kuyt, proved decisive after Liverpool's poor defending had twice allowed the home side to equalise, first through an Alessandro Diamanti penalty and then a Carlton Cole header.
Two early defeats to Tottenham Hotspur and Aston Villa have given Benitez's side no margin for error as they bid to keep pace with the early leaders.
Chelsea's ominous start of five wins from their opening five games has set a testing standard, and after failing to win the title despite losing just two league games last season, Liverpool are under no illusions about what is required over the coming months.
Three successive wins, including the Champions League victory over Debrecen, had improved the mood at Anfield and prompted Benitez to declare before this game that confidence is growing in the dressing room.
Nonetheless, the early defeats have left their mark and Carragher spoke last week about the need to embark on an unbeaten run of at least 15 games before any plaudits are handed out.
Given that background then, the last thing Liverpool needed was for the former England defender to gift West Ham the chance to open the scoring after just two minutes.
Carragher's unexpected slip presented West Ham youngster Zavon Hines with a clear route to Pepe Reina's goal and the Liverpool defender was relieved to see Hines strike his early effort against the post.
That wouldn't be the last time Hines got the better of Carragher, but it was Liverpool who responded more positively after their early shock and looked as though they would take control of the game.
The home side's cause wasn't helped by an early injury to centre back Matthew Upson, but with Torres looking in determined form, a Reds goal was always imminent.
Inevitably it came from the Spaniard, but only after he had sent an acrobatic volley wide.
He made no such mistake in the 20th minute, though, when he collected a ball from Emiliano Insua on the left hand side of the Hammers' box and simply powered past James Tomkins before rifling his right foot shot into the roof of Robert Green's net.
Despite the lead, Liverpool never looked totally comfortable with the back-line looking vulnerable to Hines' pace and Cole's power.
And it was no surprise when Hines again found his way past Carragher, this time forcing the defender into a clumsy challenge that gifted Diamanti the chance to mark his full debut with a goal and equalise from the penalty spot.
Television replays suggested the Italian's strike should not have stood as he slipped and made a double contact on the ball.
Liverpool had it all to do again, but they were given the chance to reassert control when Steven Gerrard was allowed to rise unchallenged and meet Yossi Benayoun's corner with Kuyt on hand two yards out to help the ball home.
All Liverpool needed to do was see out the remaining three minutes and reach the interval with the cushion of a lead. But once again, they showed their defensive frailties when they couldn't defend a routine corner and Cole headed home from Diamanti's cross.
With both defences all at sea, it seemed inevitable more goals would follow, but the second half proved to be a much tighter affair with Torres finally conjuring the decisive act when he met Ryan Babel's cross and headed home with fifteen minutes remaining.