Carroll was impressive in the 2-1 win.
"I thought Andy played a great game and he worked really hard," Kuyt told liverpoolfc.tv.
"I know how difficult it is to be alone up front but he battled throughout the game. He was important for us for both of our goals.
"The header for the last goal was something I had told him. I said, 'Win the header and flick it on' and he did that perfectly.
"I've been a striker all my life. Whenever there's a chance I just take it as it is. I've got a lot of belief in myself and all the players around me are always supporting me, like I try to support them.