The ball bounced off the rim, and the Rockets survived the Suns' late rally on Monday night to keep pace in the race for a postseason berth.
The Suns rallied from 14 points down in the second half to tie it in the closing minutes. Phoenix had a chance to win at the end, but Dudley missed a 3-pointer at the buzzer that would've given his team its first lead.
"That was a long 1 or 2 seconds when the ball was in the air," said Chase Budinger, who scored 15. "I'm just glad it didn't go in."
Kyle Lowry added 18 points, six rebounds and five assists for the Rockets, who stayed three games behind Memphis in the chase for the final playoff spot. Houston is now only one-half game behind Phoenix in the standings.
"We know we're close," Hayes said. "We can't really ask for help. We just have to control our own destiny, and win as many games as possible."
Aaron Brooks, acquired from Houston in a trade at the deadline, started for Nash, who sat out with a hip problem, and scored two points on 1 for 9 shooting.
The Suns acquired Brooks in exchange for point guard Goran Dragic and a first-round pick. Brooks faced his old team for the first time last week in Phoenix, a 113-110 Suns' win.
Brooks, Houston's first-round draft pick in 2007, got a loud ovation when he was introduced before the game. He pulled down the game's first rebound.
During an early timeout, a video montage of Brooks' highlights from his 3-plus seasons played on the scoreboard, prompting more applause. Brooks acknowledged the crowd with a quick wave.
"That's out of the way," Brooks said. "Having that game in Phoenix kind of helped a little bit. I'm just glad these Houston games are over, and we can go about the season."
The Suns slipped 2½ games behind the Grizzlies in the race for the No. 8 playoff spot in the West.
The Suns trailed by as many as 15, before coming back to tie it at 89-all in the final two minutes.
"It's too bad we didn't play the first half with a sense of urgency," Carter said. "We have to come out and play hard, and in the second half, we did a better job."
The Suns missed 9 of their first 14 shots and Houston built a double-digit lead in the first quarter. Brooks started 0 for 4, including an uncontested 3-point try near the 2-minute mark of the quarter.
But in the fourth quarter, the Rockets were the ones who started cold, missing 12 of their first 15 shots.
Childress swished a 3 from the corner, and Grant Hill hit a mid-range jumper to draw the Suns within one, at 87-86, with three minutes left. Hayes hit two free throws with 2:17 left, but Dowdell hit a 3 from the corner to tie it.
Lowry sank a jumper from the wing, but Hayes missed two free throws with 30 seconds left to keep the score 93-91.
Carter missed a 3-pointer, Hayes rebounded and was fouled by Hill with 24 seconds to go. Hayes made his second free-throw attempt for a 94-91 Houston lead.
"To be honest, all four of them, the last four that I shot," Hayes said, "they all felt good."
Marcin Gortat got free for an easy dunk with 16 seconds left to cut Houston's lead to 94-93. Budinger was fouled by Brooks and hit a free throw with 11 seconds to go.
Carter had the ball as the final seconds ticked away, and had to pass to Dudley out of a double-team for the final shot.
Houston forward Luis Scola sat out for the third straight game (sore left knee). ... The Rockets failed to reach 100 points for only the second time in 16 games. ... Nash was officially listed as being out with "pelvic instability." Houston G Terrence Williams sat out with "lower abdominal distress." ... The Suns lost for only the third time in their last nine road games. ... Phoenix had scored at least 110 points in its last eight games against the Rockets.