Rosenfels, 31, agreed to a two-year, $9 million extension with the Vikings, who gave up a fourth-round draft choice to acquire him, a source told ESPN.com's John Clayton. Rosenfels had one year remaining on his deal with the Texans that would have paid him $1.35 million.
He flew to Minnesota on Thursday and was scheduled to undergo a physical Friday morning.
Rosenfels' agent, Rick Smith, told The Associated Press his client "is extremely excited for this opportunity."
Rosenfels is expected to compete with incumbent Tarvaris Jackson for the starting role in Minnesota.
It's the second season in a row that the Vikings have tried to trade for Rosenfels. Last year, the two teams could not agree on compensation, so Minnesota brought in veteran Gus Frerotte as a backup.
"It's something we've been working on for a while and it finally came to fruition," Smith said. "This is the first time in his career that he's had a great opportunity to start. Nothing has been promised to him. He's just extremely excited for this opportunity."
The addition of Rosenfels meant there was no room for Frerotte, who was waived Friday. Frerotte expressed no surprise or anger in the Vikings' move.
"I talked to Coach [Brad] Childress and he started to explain the whole deal," Frerotte told the Minneapolis-St. Paul media outlets. "I stopped him and said, 'You don't have to explain.' I appreciate you having me come up there and believing in me. I had a lot of fun and am proud of everything we accomplished as a team. I really enjoyed playing with everybody on the Vikings.
"I just wanted to thank [Childress] and Mr. Wilf for letting me be a part of the team. Most of all I wanted to thank all Vikings fans. Minnesota is one of my favorite places to play and the stadium is awesome. I really enjoyed myself."
Rosenfels went 2-3 in five starts for the injured Matt Schaub in Houston last season. He threw for 1,431 yards with six touchdowns, 10 interceptions and a 66.7 completion percentage.
Bringing in someone to compete with Jackson for the starting job was one of Minnesota's biggest offseason priorities after losing to Philadelphia in the playoffs.
Jackson was benched after two losses to open last season and Frerotte went 8-3 in his place.
When Frerotte went down with a back injury, Jackson flourished to finish the regular season. He rallied the Vikings to a victory over the Lions after Frerotte got injured and won NFC Offensive Player of the Week honors the following week after throwing four touchdown passes in a road victory over Arizona.
In the first round of the playoffs, however, Jackson took a big step back against the frenetic blitzes of the Philly defense. He went 15-for-34 for 164 yards and had an interception returned for a touchdown in a 26-14 loss to the Eagles that ended Minnesota's season.
"We need to put somebody in place that will push [Jackson] and compete with him, and I think competition is the nature of the game," Childress told reporters at the NFL combine last week in Indianapolis. "At some places it may not be ... [but] in our situation, we need to have a good, healthy competition because I think that makes everybody better."
John Clayton is a senior NFL writer for ESPN.com. Information from The Associated Press was used in this report.