Lee, 36, is hitting .287 with five homers, 29 RBI, 15 doubles, a triple and 24 runs scored over 258 at-bats in 66 games this season.
He spent five-plus seasons with the Astros after signing a six-year, $100 million contract as a free agent on Nov. 24, 2006.
According to ESPN.com, the Astros will absorb the remainder of Lee's $9 million for this season, minus a prorated salary of roughly $250,000.
The Marlins are hoping that Lee will offer an instant upgrade at first base as they attempt to get back in the NL East race.
Miami entered Wednesday as one of the worst ballclubs in Major League Baseball in terms of offensive production at first base.
Lee previously played under the guidance of Marlins manager Ozzie Guillen as a member of the Chicago White Sox in 2004.
For the Astros, it was time to move the veteran slugger in order to clear a path for the purpose of giving younger players a look.
"Carlos has been an important part of our team and our community in Houston for almost six seasons, and he will be missed," Luhnow said in a statement. "We made this move with an eye toward the future and are very excited about adding Dominguez and Rasmussen to our talent base. Both players have a bright future."
The Astros made it known Wednesday that Dominguez will join the major league club Thursday in the four-game series finale in Pittsburgh.
Rasmussen will report to Double-A Corpus Christi.
Dominguez, 22, was selected by Miami out of Chatsworth (Calif.) High School in the first round of the 2007 First-Year Player Draft.
He was hitting .234 with seven homers, 46 RBI and 14 doubles over 286 at-bats in 78 games at Triple-A New Orleans this season.
Dominguez entered 2012 ranked as the Marlins' No. 4 prospect by Baseball America after rating as the club's top prospect in 2011.
The young infielder made his major league debut with Miami last season as a September call-up and ultimately hit .244 with two RBI, four doubles and two runs scored over 45 at-bats in 17 games.
Rasmussen, 23, has compiled a 4-7 record with a 3.90 ERA over 87 2/3 innings in 16 starts for Class A Jupiter this season.
He had been recently promoted to Double-A Jacksonville.
The young left-hander was selected by the Marlins out of UCLA in the second round (73rd overall) of the 2010 First-Year Player Draft.
Rasmussen entered 2012 ranked as the Marlins' No. 7 prospect.