Fellaini headed home a 90th minute winner at Bolton last night, just days after doing something similar in the draw with Manchester United.
On breaking the Club-record fee to land the Belgian, Moyes was keen to point out Fellaini would take time to settle.
But having notched three important goals in his last four outings, Moyes says the former Standard Liege man is rapidly becoming a stalwart of the side.
"He's getting us a few goals," he told evertonTV. "He's actually stepping up to the mark where we had thought we would nurture him in slowly. But he's going to become a big man for us and a big figure. For a 20-year-old boy I think he's doing terrifically well."