The Rockets led by five early the first extra period, but the Timberwolves rallied and went ahead 98-96 on Ellington's 3-pointer. Brooks hit two free throws with 1:25 left to tie it again, and both teams missed potential winning shots in the last minute.
Jefferson hit a pair of free throws and a hook shot to give Minnesota a 102-98 lead early in the second overtime. Ariza hit a jumper from the wing, then converted a three-point play.
Ellington lost the ball at midcourt with 16.9 seconds left, Brooks' free throws put Houston up 105-102, but Ellington tied it again with a 3-pointer with 8.4 seconds left.
"I am very proud of the way we fought back," Minnesota coach Kurt Rambis said. "This is a great example of what happens when you keep playing the game."
Brooks swished a 3-pointer in the first minute of the third extra period, but Hayes fouled out when he bumped Jefferson on a layup. Carl Landry's turnaround gave the Rockets a 110-107 lead before Jefferson answered again.
Ariza sank two free throws, Gomes was called for an offensive foul and Battier sank a 3-pointer from the corner with 1:55 left to give the Rockets a 115-109 lead. Houston hit enough free throws at the end to escape with its eighth straight home victory.
"When you get into overtime like that, sometimes the way the ball bounces could determine the game," Brooks said. "The ball bounced our way this time."
The Rockets seemed to have the victory in hand in the fourth quarter after Brooks' reverse layup with 46 seconds left put Houston up 87-82.
Jefferson scored inside and Brooks and Landry missed free throws in the final 10 seconds, allowing Minnesota a chance to tie. Kevin Love threw a perfect inbound pass to Brewer, who heaved a mid-court shot that swished in as the buzzer sounded.
"I was right behind Corey Brewer in front of their bench, and it looked good," Hayes said. "They went nuts."
The Rockets should've never let it come to that.
They mounted an early 17-0 run, delivered 12 assists on 14 baskets in the quarter and built a 30-13 lead.
The Timberwolves cut the deficit to nine by halftime, then opened the second half with an 11-3 burst. The Rockets missed nine of their first 10 shots after the break, including all four from 3-point range.
Houston then scored on five consecutive possessions, capped by Battier's 3-pointer with 3:11 left in the quarter to push the lead back to 10.
But the Rockets missed their first six shots of the fourth quarter as Minnesota crept closer. Jefferson's hook shot with 8:43 left cut Houston's lead to 69-68 before Brooks hit a 3-pointer.
A few minutes later, Damien Wilkins sank a 3-pointer and Jefferson flipped in a short jumper to give Minnesota a 78-77 lead. Hayes scored in the lane and Brooks swished another 3-pointer with 3:51 remaining to put the Rockets back in front.
The Rockets improved to 3-0 in triple-overtime games. It was their first since a 123-121 win over Atlanta on Feb. 22, 2004. ... Timberwolves C Ryan Hollins missed his fifth straight game with an upper respiratory infection. ... Minnesota's first-quarter point total matched a season low for a Houston opponent in any quarter this season.