By halftime, Brown had missed from 40 and 53 yards, Hasselbeck had thrown for just 39 yards and Alexander had run for only 15 yards.
"I told the team at halftime I knew we would win the game, that somehow, some way, we would win the game," Holmgren said. "And we did."
Brown made all three of his kicks in the fourth quarter, but didn't forget that he missed from 40 and 53 yards in the first half.
"He pointed out that if he had made the earlier ones, maybe we wouldn't have been in that position at the end," Holmgren said.
Or if Alexander had averaged more than 1.7 yards per carry. Or Hasselbeck, bothered by a sore left hand and balky right knee, had completed more than six of his 14 pass attempts.
Denver (7-5) had five turnovers, its most at home since 2000.
"That's all Brandon Marshall," Cutler said. "I didn't do anything. I just got him the ball and he made a big play for us. We have a lot of guys like that and we need to spread the ball around more."
Before that, Cutler had thrown for just 72 yards.
On third-and-4 from his 20-yard line, Hasselbeck found Nate Burleson for 16 yards. Brown, who was good from 44, 23 earlier in the fourth quarter, lined up from 50 yards and booted it right down the middle.
Cutler, the last of the "Big Three" quarterbacks from the 2006 draft class to make his debut, met the same fate as Vince Young and Matt Leinart -- a loss. Cutler finished 10-for-21 for 143 yards with two TDs and two interceptions.
"To have a game like this and have two picks and have one housed, it's tough," Cutler said.
"I think he did pretty good for being his first game," Shanahan said. "I'm a little disappointed in myself. I think I put too much in the game plan for this being his first game."
Denver lost middle linebacker Al Wilson to a sprained neck in the fourth quarter -- Wilson was carted off the field.
Seattle's only score in the first half came when Cutler was wrapped up by Bryce Fisher and threw the ball up for grabs. Tapp snared it and sprinted 25 yards for the score that cut Denver's lead to 10-7.
"I made some mistakes," Cutler said. "I blame myself."
The Broncos hadn't lost three straight since 2003 and hadn't dropped three in a row at home since 2002.
Hasselbeck was just 10-for-20 for 75 yards when the Seahawks got the ball at their 39 with 9:42 left, trailing 13-7. He hit Darrell Jackson for gains of 27 and 33 yards, then handed to Alexander, who took it in from 1 yard to give Seattle a 14-13 lead.
Brown kicked field goals of 44 and 23 yards after Denver turnovers to make it 20-13 before Marshall's sensational TD.
Hasselbeck finished 16-for-28 for 168 yards.
Cutler, the first rookie to start for Denver since Tommy Maddox filled in for an injured John Elway 14 years ago, expected to spend this season watching and learning. But the Broncos turned to the former Vanderbilt star and 11th pick in the NFL draft after an ineffective Jake Plummer was benched following back-to-back losses to AFC West rivals San Diego and Kansas City that threatened to derail their season.
Denver's Tatum Bell, who missed three of the last four games with sprained toes, gained 133 yards on 23 carries but had just 21 yards in the second half.
Alexander, coming off a 201-yard performance against the Packers in his return from a two-month absence with a broken foot, ran 26 times for 90 yards despite the loss for much of the game of his lead back, Mack Strong, to a sprained ankle.
The Broncos took a 13-7 lead into the locker room when Jason Elam kicked a 41-yard field goal with a second left after injuring his left hamstring while gaining 2 yards on a fake field goal on fourth down.