The Frenchman scored twice from the penalty spot and added a stunning third late on for his first treble for City as supporters went home in fine voice.
Yet, at half time a 4-1 scoreline looked a distinct improbability as City, who had failed to win in their first two home fixtures, fell behind to a Juan Pablo Angel goal.
But it was an altogether a different story in the second half as City raised the tempo and destroyed Villa with some scintillating play.
Manager Kevin Keegan was fulsome in his praise of Anelka, especially for his perseverance after a frustrating first 45 minutes.
He said: "In the first half there were three or four times when Nicolas was in good situations, but he didn't hit the target.
"But like all great strikers, and I think he is a great striker, he kept going and was rewarded in the second half."
Keegan added that he believed Anelka, who has already scored six goals, can still improve further.
He explained: "Nicolas seems to have been around a long time, but he is just 24.
"I was 27 before I learned what football was all about. If he can learn in the next three years some of the things he can improve on then by 27, when he should be at his peak, he will be just about the best in the world."
Villa stunned City taking a 31st minute lead through a close-range header from Angel.
And City were lucky not to be further behind as Marcus Allback and Angel wasted two further glorious chances.
City came out with far more purpose in the second half and within five minutes of the restart were 2-1 ahead.
New boy Steve McManaman burst into the penalty area only to see his progress thwarted by the hand of Mark Delaney and Anelka scored his first from the penalty spot.
German Michael Tarnat thundered home another of his speciality free-kicks as City took control.
Anelka was spot on again midway through the half after Sun Jihai was tripped in the penalty area by Peter Whittingham.
Anelka completed his hat-trick seven minutes from the end with a terrific drive from outside the penalty area.
Villa manager David O'Leary had no complaints about his side's defeat, his main grumble being that his side failed to convert more of the chances they created.
He said, "We will create chances, but at the moment we aren't taking enough of them.
"We played well in the first half, but cannot afford to go out and play like we did in the second."