Carroll delivered a crushing blow to Manchester City’s Champions League ambitions by taking his frustration out on Mancini and his expensively-assembled players with his first two goals for Liverpool.
Dirk Kuyt also scored as Dalglish inspired his side to another impressive victory over one of the big four.
Yet this was more impressive than the 1-0 win at Chelsea or the 3-1 victory against Manchester United.
As good as Liverpool were, City were awful. Their performance was summed up when substitute Mario Balotelli was subbed himself after a woeful display.
Balotelli had made an earlier-than-expected appearance for Tevez, the Manchester City captain limping off with a hamstring injury which is likely to rule him out of Saturday’s FA Cup semi-final with neighbours United at Wembley.
“I don’t think he will play,” Mancini said afterwards. “We will try to recover him, but we also think of the other six league games.”
As for his side’s alarming performance, Mancini blamed himself rather than his under-performing players.
“I’m disappointed with myself because I think I made a mistake,” added the City manager. “It was my fault.
Not with the players because the players played 100%. Probably I made a mistake in the last two days but I know why. I did three changes and brought in three fresh players. Our focus was not good.”
City’s performance was as jumbled as Mancini’s words. They have now failed to win any of their last six Premier League away games.
City are only three points ahead of fifth-placed Tottenham, who they entertain on May 10, while their London rivals also have a game in hand.
City had arrived at Anfield boasting more clean sheets than any other team in the Premier League. Yet hopes of a 15th were undone by a shambolic first half defensive display.
City had already been handed a seventh minute warning of Liverpool’s rampant mood by Luis Suarez, the Uruguay forward’s powerful low shot striking the post after the faintest of touches by keeper Joe Hart.
But Hart was unable to keep out England team-mate Carroll’s majestic finish six minutes later.
It came after Tevez had originally lost possession, Raul Meireles’ ambitious long-range effort cannoning off Vincent Kompany straight to Carroll, whose left-foot 25-yard finish flew past Hart into the net.
Having arrived at Liverpool injured, Carroll has endured a long and frustrating battle to get fit. But this was pay back time.
No sooner had City fallen behind than Tevez threw off the captain’s armband after damaging his hamstring before disappearing down the tunnel.
Then Mancini saw his players capitulate as City went from one goal down to trail 3-0 in the space of 60 seconds.
Soon after Fabio Aurelio had been fortunate to escape a red card after an ugly two-footed challenge on Adam Johnson, Liverpool doubled their lead in the 34th minute through Kuyt’s well-executed low finish.
The inquest was still going on in the City defence when Carroll got the better of Aleksandar Kolarov to head Meireles’ cross beyond Hart.
The only consolation for the lacklustre visitors in the second half was that they did not concede again.
Liverpool were without Steven Gerrard, Daniel Agger, Glen Johnson and Martin Kelly, and gave a debut to 18-year-old right-back John Flanagan.
The teenager played his part in a memorable result which lifts Dalglish’s side to within five points of Tottenham and keeps alive their hopes of qualifying for next season’s Europa League.
“Mathematically it’s possible, whether we get there or not remains to be seen,” said Dalglish, who was delighted as Carroll got off the mark.
“Andy’s first one was a fantastic strike, the second a good header. The more Andy and Luis Suarez play together, the better the understanding.”
Liverpool (4-2-3-1): Reina 7 Flanagan 7, Carragher 7, Skrtel 7, Aurelio 6,; Lucas 7, Spearing 7; Kuyt 8 Meireles 7, Suarez 7, Carroll 9 (Ngog 90).
Manchester City (4-3-3): Hart 6, Boyata 5 Kompany 5, Lescott 4, Kolarov 5; Y Toure 5, Barry 5, Milner 4 (Silva 59, 5); A Johnson 6, Dzeko 5, Tevez 5 (Balotelli 16, 4, De Jong 83, 3).
Referee: Mark Halsey 6.