Burnett, 46, proposed to Downey, 46, during a Thanksgiving trip to Mexico, publicist Jim Dowd said.
"They have never been happier," Dowd said. "They are enjoying this exciting time with their children and their families."
Both have children from previous marriages. Burnett has two sons, ages 13 and 9, and Downey has a 10-year-old daughter.
The couple, who started dating in 2004, plan to marry next year.
"I have found my angel," Burnett said in a statement.
Burnett is the producer of reality shows including "Survivor" and "The Apprentice." Downey starred in the TV show "Touched by an Angel" from 1994 to 2003.