The 22-year old striker scored his first goals for his new club as they cruised to a comfortable win at Anfield - his performance endearing him to the home fans.
The former Newcastle United star struck a sweet shot from distance in the 13th minute before grabbing a second, and the side's third after Dirk Kuyt struck, with a superb 35th minute header to essentially finish City off.
Carroll said it was a great feeling to open his account for Liverpool and was delighted to be heavily involved in the win.
"It's been a great day for us, getting my first goal and a great result all round," Carroll told Sky Sports 1.
"The reception I've got from the fans has been great, it's been fantastic since I got here."
On his first goal he added: "It dropped nicely for me on the edge of the area and I knew I'd hit it (cleanly), and I hit it sweeter than thought.
"We went out to put pressure on them and that's what we did and got goals early."