Steve Nash had 22 points and 18 assists but the Suns lost for the second time in eight games.
Cousins, a rookie 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.
"We've been coming so close against good teams and getting turned away in the fourth quarter," Sacramento coach Paul Westphal said. "Tonight we pushed through and beat a good team on the road, especially with all the turmoil, or whatever you want to call it."
Phoenix, which had won 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, blowing a chance to climb two games above .500 for the first time all season.
"It's one of those losses that you are going to look back at and it's going to hurt," coach Alvin Gentry said.
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. Instead, the ball went to 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 away from home -- over Portland, the Los Angeles Lakers and Phoenix.
The Kings have not said whether Cousins would be banished for more than one game.
"We missed Cuz tonight but we couldn't dwell on it," Greene said. "We had to come out and still play well."
There were no hard feelings, Greene said.
"Things happen," he said. "Everybody is not perfect. We try not to let things off the court distract us on the court. It is life. Everybody is not perfect and we've got to move on."
Landry, 0 for 6 from the field in the first half, was 5 of 7 for 12 points in the third quarter. Greene shot 4 of 5 in the fourth quarter to finish 7 of 10.
Nash was fouled on a 3-point attempt for the second time in the fourth quarter, but the best free throw shooter in NBA history missed one of 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.
"They just kept scoring and kept scoring and we couldn't stop them," Gortat said. "And when we didn't stop them, we couldn't execute. So, I would say it's obviously a really tough loss which cost us a lot."
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. Darnell Jackson's 3-pointer just before the halftime buzzer cut the Suns' lead to 57-50 at the break.
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 to 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.