The victory came in the opener of their best-of-seven Western Conference quarterfinal series.
Denver held a 44-42 halftime lead, and 67-66 entering the fourth quarter.
Denver was able to pull away after San Antonio had taken a 75-72 lead with 5:28 left in the game.
Iverson had 31 points and five assists, and Anthony had 30 points and eight rebounds.
Nene Hilario added 13 points and 12 rebounds for the Nuggets, who won 10 of their final 11 regular-season games.
Tony Parker led San Antonio with 19 points, eight assists and six rebounds.
Game 2 is set for Wednesday night in San Antonio.