Koeman has introduced a no-nonsense regime at Everton since replacing Roberto Martinez as boss at the start of the summer.
Lukaku said: “I had a feeling from the first day we met he would improve us as a team and especially me as a player.
“I knew it was coming. I had a good performance with Belgium and then I knew I was back in business.
“I could have had five or six but I'm happy with three. It's my first hat-trick for the club in the Premier League so I'm delighted.
“I knew fitness-wise I was not at my best in the first weeks of the season because I came in later than the others.
“I had to work twice as hard and now it's paying off so I am really happy."