Carl Landry added 20 points for the Kings, all but two in the second half. Samuel Dalembert, starting in place of Cousins, had 18 points and 15 rebounds.
Steve Nash had 22 points and 18 assists but the Suns lost for just the second time in eight games.
The rookie Cousins, averaging just under 14 points a game, did not make the trip because of an altercation in the locker room with Greene after the Kings' 99-97 home loss to Oklahoma City on Saturday night.
Tyreke Evans scored 21 for Sacramento. Marcin Gortat had 20 points and 12 rebounds for the Suns.
Phoenix, winner of three in a row, led by as many as 10 in the first half and was up 81-78 after three quarters, but the Kings opened the fourth with a 15-3 run. Greene scored the last five in what turned out to be the decisive surge. His 3-pointer put Sacramento ahead 91-83, then Nash lost the ball out of bounds and Beno Udrih found Greene inside for a layup that made it a 10-point lead with 6:03 to play.
The Suns never got closer than four again.
The Kings said in a brief statement before the game that Cousins would not play "while the organization reviews the situation that occurred in the locker room following last night's game versus Oklahoma City."
Cousins reportedly was upset that Greene did not pass him the ball for the final shot in an 89-87 loss. Instead, the ball went to Tyreke Evans, who missed a 3-pointer.
All five of the losses in the Sacramento skid came at home, none by more than five points. The Kings are 6-16 on the road but have won their last three straight away from home - over Portland, the Los Angeles Lakers and Phoenix.
Landry, 0 for 6 from the field in the first half, was 5 of 7 for 12 points in the third quarter. Greene made 4 of 5 shots in the fourth quarter to finish the game 7 of 10.
Nash was fouled on a 3-point attempt for the second time of the fourth quarter, but the best free throw shooter in NBA history missed one of the three from the line and Sacramento led 109-104 with 27.7 seconds left. Two free throws apiece by Dalembert and Udrih sealed the victory.
The Suns used a 22-4 outburst to take a 29-21 lead in the first quarter. Landry's steal and subsequent two free throws - his only points of the first half - cut Phoenix's lead to 48-47. Channing Frye scored the first five in a 9-0 surge that ended on Robin Lopez's steal and fast-break dunk to make it 57-47. Darnelll Jackson's 3-pointer just before the halftime buzzer cut the Suns' lead to 57-50 at the break.
Notes: The Kings played the first of seven in a row on the road, with the All-Star break coming after the first three. ... Phoenix had won 12 consecutive Sunday home games, a streak dating from Feb. 22, 2009. ... After having no dunks all season, the Suns' Jared Dudley got his second in the last three games. ... Sacramento had not won since a home victory over New Orleans on Jan. 29.