The Ducks sent Carney, with them since 2001, and defenseman Juha Alen to Vancouver for minor-league defenseman Brett Skinner and a second-round pick in this year's NHL entry draft.
The Ducks got Friesen from Washington, giving the Capitals a second-round pick this year. O'Donnell came in a trade with Phoenix for minor-league center Joel Perrault.
Ducks general manager Brian Burke said the moves just before the trade deadline were made to help the team make the playoffs this year and also with an eye to the future.
"Our goal going into the trade deadline was to reward the group that has put together this run and add sensibly to that," Burke said in a conference call. "From our perspective, we believe we upgraded.
"We kept our kids, with the exception of Joel Perrault, and added some depth in draft picks," he said.
After struggling earlier this season, the Ducks have been playing well and are third in the Pacific Division and battling for a playoff spot.
Carney is a veteran of 14 NHL seasons and has 198 points and 760 penalty minutes. He played 271 of his 859 career games for Anaheim.
Alen has been playing in Finland.
The 29-year-old Friesen has 500 points in 803 NHL games, including seven points in 33 games for Washington this season.
O'Donnell, 34, had eight points in 57 games for the Coyotes this season and 148 points and 1,356 penalty minutes in 750 NHL games.