Anthony topped all scorers with 30 points and hauled in a team-high 10 rebounds, while long-range shooter Steve Novak had 17 points in New York's season opener.
LeBron James scored 23 points to pace Miami, who dropped to 1-1 after an opening win against the Boston Celtics.
It was the Knicks' first home game since last season's opening-round play-off series against the Heat, which they lost in five games. The Heat also beat New York three times during the regular season, so the Knicks had something to prove.
"You always want to have good game, now matter what," said Knicks point guard Raymond, who scored 14 points and had a game-high nine assists. "But it's always good to beat the defending champs."
After a moment of silence to show support for victims of the storm, the Knicks came out firing, relying on their three-point sharp-shooting and some defensive tenacity to take quick command in what the New Yorkers viewed as a measuring stick game.
New York used a 10-0 run from 7-6 and later in the first quarter scored 11 points in a row to turn a 19-12 advantage into a 30-12 cushion to the delight of the roaring Garden crowd.
The Knicks led 33-17 after a first quarter in which they shot 60 per cent from the floor, including a red-hot, 5-of-8 from beyond the arc as Anthony notched 16 points.
Miami cut the lead down to 11 at 55-44 by half-time but New York came out with renewed fire in the third quarter, extending the lead to 23 points, boosted by two three-pointers from crowd favourite Novak and the Knicks were never seriously challenged.
New York shot a blistering 19-for-36 from three-point range as the penetration of their point guards created wide open opportunities from long range.
The Knicks were supposed to open their season against the Nets on Thursday in their new home in Brooklyn, but the game was postponed because of the crippled state of the city's mass transit system.
"We were kind of excited about the start of the season after missing last night," said Anthony. "We were excited to come out on the basketball court and see where we were at as a team."
Anthony took a microphone before the game and on behalf of the players thanked the fans for turning out after all the hardships caused by the storm that hit the area on Monday.
"We gave them a good show tonight," he said after the victory. "That was the least we could do."
Dwyane Wade, who scored 15 points for Miami on 7-for-10 shooting, said the Knicks rose to the occasion.
"They played with a lot of energy, they played with the crowd behind them, a lot of emotion," Wade said.
"They played a very good game, the kind of game you expected them to play. We just wish we had played a little better."
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