Stoudemire scored 23 points, Steve Nash had 16 assists and the Suns recovered from a slow start to beat the Rockets 111-105 on Tuesday night.
Jason Richardson scored 20 points for the Suns, who have won eight of their last 11 games in Houston.
The Suns held the Rockets to 32 percent shooting (9 of 28) in the fourth quarter and outscored Houston 14-8 in the final 3 minutes to pull out the win.
"We've got a little toughness," Nash said. "We're not physically the toughest group, but mentally we're tough. We started to battle at the end of the game and we got a lot of stops."
"We started missing shots and they started making shots," said Brooks, who scored 16. "We were in a position to win the game at the end. They made the plays and we didn't."
The Suns (10-2) rallied from a double-digit deficit to win for the fifth time this season. They're the only team to reach 100 points in all of their games so far.
Nash didn't score in the first half and went 2 for 10 from the field in the game. He went 8 for 8 from the free-throw line in the fourth quarter and finished with 12 points. All five Phoenix starters reached double figures.
"We had a really good team effort," Nash said. "It was a bit of a struggle tonight, we weren't great. But we got some really good contributions from a lot of people."
Richardson's 3-pointer with 8:11 left in the third quarter gave the Suns their first lead since the opening minutes. Brooks hit a shot from the baseline before Nash made a pull-up jumper for his first field goal, giving Phoenix a 63-62 lead.
The pace of the game picked up and the lead changed hands eight more times over the next 6 minutes. Landry dunked and Luis Scola sank a baseline jumper in the last minute of the quarter to give Houston an 82-78 advantage.
Nash returned from a short rest with 9 minutes left in the game and the Rockets leading 85-84. He immediately found Leandro Barbosa cutting inside for a layup, and the Suns moved back in front.
The teams traded baskets into the final minutes before Nash delivered his 15th assist to Richardson for a layup with 2:59 left that put the Suns up 100-98.
Brooks missed a short jumper and reserve forward Channing Frye sank a 3-pointer from the top of the key with 1:51 remaining that made it 103-98. Landry split two free throws and backup center Jarron Collins converted a three-point play to put it out of reach.
Suns reserves scored 34 points.
"The second unit, the guys off the bench, do a great job of keeping themselves in shape," Stoudemire said. "It's great to see those guys come in and contribute the way they do."
"We are still in the emerging stages," forward Shane Battier said. "We are trying to figure out what's good for us, especially down the stretch. That's something we are going to have to go through and look at this game, learn from our mistakes in the fourth quarter and grow."
The Rockets finished 9 for 26 from 3-point range, after hitting five of their first seven attempts to take a 21-9 lead. Houston (6-5) also outrebounded Phoenix 14-5 in the opening quarter and led 34-22.
The Rockets have reached 100 points in 10 straight games, the team's longest streak since a 12-game run in 1995-96. ... Phoenix is 10-2 for the third time in six seasons. ... The Suns' 22 first-quarter points were a season low. ... Hall of Famer and former Rockets C Hakeem Olajuwon had a courtside seat. He shook hands with referee Joe Crawford during a timeout in the first quarter.