The Suns opened the final period of their season opener with a 9-2 run that overcame a three-point deficit.
Stoudemire and Nash then combined to score seven straight points down the stretch to give Phoenix a 10-point advantage with 42 seconds to play.
Stoudemire scored 23 points, while Nash had 18 points and nine assists. Grant Hill scored 13 in his Phoenix debut.
Seattle rookie Kevin Durant had 27 points and six rebounds.
The Suns scored the last six points of the first half to close within three at the intermission.
Phoenix won its season opener for just the second time in nine years.