Anthony scored 17 of his game-high 36 points in the second half, including a critical steal and lay-up with two and a half minutes remaining.
Andre Miller's basket with just over a second left enabled the Denver Nuggets to edge the Golden State Warriors 97-95 at Pepsi Center in their opener - the 37-year-old scored 18 of his 28 points in the fourth quarter.
A 60-35 first half paved the way for the Brooklyn Nets to open their series against the Chicago Bulls with a 106-89 victory at Barclays Center.
Deron Williams led the way for the Nets with 22 points but Carlos Boozer was the game's top scorer with 25 for the visitors.
A 37-22 final quarter was the icing on the cake for the Los Angeles Clippers, who sunk the Memphis Grizzlies 112-91 at Staples Center.