The Sounders would have set an MLS record if they had beaten the Rapids an eighth straight time.
Drew Moor tapped in a goal off a corner from Nick LaBrocca to give the Rapids (8-7-7) a 1-0 lead in the 60th minute. Moor got in front of defender Jhon Kennedy Hurtado for the open shot in front of the net.
"I think I probably stole one from Nick there," Moor said. "It was one of those plays that I had a man marking me on the previous corners and the one just before the goal nobody really marked me.
"It was a perfect ball from Nick and I put it away just to be safe. I think I beat the ball to the back of the net actually."
Colorado had confidence it could score off a corner kick.
"We knew we would get some chances in the game like that one," said Rapids coach Oscar Pareja about Moor's goal.
"Unfortunately, we conceded the goal that they got. But it was a good match. It was a fight and I like the way we responded."
Seattle coach Sigi Schmidt said his team should have stopped Colorado's goal.
"It was a lost assignment on the corner," Schmidt said. "The goalkeeper (Andrew Weber) came fishing as well, but we completely lost the man we were marking."
Seattle striker Eddie Johnson (four goals) missed the game because he is with the United States for its Gold Cup quarterfinal match versus El Salvador on Sunday.
Nine other Sounders are injured, including regular goalkeeper Michael Gspurning, who did not play the second half of a 1-0 loss to San Jose on July 13 due to an arm injury.
Yedlin got his chance after Lamar Neagle blasted a shot off the far post. Yedlin kicked a shot that goalkeeper Clint Irwin blocked right back to Yedlin for his header.
"A little bit lucky," Yedlin said about his goal. "Lamar had a good shot, but it was unlucky to go off the post. It bounced to me, I hit it first and the keeper made a good save and I was lucky it bounced back and I headed it."
Yedlin, 20, became the youngest player in club history to score in league play.
Moor, a defender, said the Rapids needed to do a better job on defense to stop the goal by Yedlin.
"We need to clear the ball further," Moor said. "I had a couple of chances to clear. We got into a mode where we were just trying to block shots and Yedlin got a couple of efforts at it. We were really pinned in for about two minutes before the ball went into the goal."
Colorado's Hendry Thomas blasted a shot off a free kick past the defensive wall in the 81st minute, but goalkeeper Andrew Weber went to his left to tip the ball away.
Seattle (7-7-4) had lost three of its past four. A victory would have given the Sounders their first five-game home winning streak.
The Rapids are starting to gain confidence.
"We've got to play like we did today," Moor said. "It wasn't pretty today, but it was a short week for us and it was a great fight."