The Goodison Park chief has won the Manager of the Month, while the Scottish forward has picked up the player's trophy after scoring all of his new club's goals this season
The Toffees currently sit fourth in the table after taking seven points out of a possible 12 last month - with wins against Swansea City and Southampton their standout results.
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Moyes told the club's official website: "With the quality of the Premier League it is really hard to qualify for the Champions League but I wouldn't say we can't do it. It is possible and you have to aim for that.
"If we fall short and end up in the Europa League then so be it, but in recent years we have been pretty close, finishing sixth, seventh or eighth, so we want to do that at the very least."
Meanwhile, Fletcher's five goals in four games helped the Black Cats to remain unbeaten for the month, and the Scotland international is delighted with his award.
"I'm obviously pleased with the award but results are more important," he told the club's official website.
"To stay unbeaten in September was a real positive for the lads and I'm pleased I grabbed a few goals as well."