Kompany praised his team for their patience in grinding out the win.
He told Sky Sports 2: "We have never had an easy game against Everton. I thought defensively we looked good today and we're really happy with the performance.
"We didn't let ourselves get surprised. Sometimes it's just a matter of patience and the goals will come."
He continued: "We have played maybe one bad 45 minutes in the whole of the league - against Fulham away. Today we didn't maybe impress but we did what we had to do."
The Belgium defender had special words for man of the match David Silva, who hit the post, had a goal disallowed for offside and set up the clinching goal for Milner with a brilliant through-ball.
Kompany said of the Spain midfielder: "He's nowhere near as good on the pitch as he is in the dressing room. He's a great character to have around the team."