Benitez met with Gillett and co-owner Tom Hicks hours after Sunday's 1-0 defeat by Manchester United at Anfield.
Gillett told the Liverpool Echo: "Rafa is the one we want as manager further on, and we have faith in him.
"I've been married over 40 years and have first-hand knowledge about how to solve an argument."
Gillett, who was speaking at a meeting of the Liverpool Former Players' Association, added: "This wasn't supposed to happen, but it has happened, and now the focus has to be on finding a common platform to continue our co-operation."
However, the American hinted that a decision has yet to be made on Liverpool's transfer policy in the January transfer window.
"The transfer window opens soon and we might use the option that gives us, or we might not," Gillett added.
"Liverpool is a great club. We are well aware that we have something very valuable in our hands."