Edmonton acquires Minnesota's fourth-round pick in the upcoming draft this summer.
Bryzgalov has played just 20 games for the Oilers after being signed by Edmonton following a disastrous first month of the season.
The Russian goalie, who turns 34 in June, was looking for work after Philadelphia bought out seven of the nine years left on the contract he signed in 201, under which the Flyers will pay him $23 million US over 14 years.
The Oilers signed him Nov. 8 to a one-year, $1.75-million deal in which $ 1 million was a signing bonus.
Bryzgalov has a .908 save percentage and 3.01 goals-against average in 20 games this season, with one shutout, the 31st of his NHL career.
Bryzgalov began his career in Anaheim and was put in a position to sign the mammoth deal in Philadelphia after four stellar seasons in Phoenix.
He has a .913 save percentage and 2.57 average in 445 regular-season games, and won four playoff contests during Anaheim's 2007 Stanley Cup run.
The move comes one day after the Oilers announced a two-year deal beginning next season for goalie Ben Scrivens.
Reports out of Anaheim suggest that Edmonton could be eyeing goalie Viktor Fasth.
The 31-year-old Swede surprised in his first NHL season in 2013 with the Ducks. Fasth posted a 15-6-2 record, four shutouts and a .921 save percentage, with a 2.18 goals-against average.
But he's appeared in just five games for Anaheim this season as Jonas Hiller has starred and Frederik Anderson has emerged as backup.
Minnesota's goaltending situation has been in flux. Veteran Niklas Backstrom is again injured, while Josh Harding starred early this season but has dealt with symptoms of multiple sclerosis.
Saskatoon native Darcy Kuemper has starred of late for the Wild, but had just six games of NHL experience heading into this season.