The 48-year-old becomes the 10th bench boss in TFC's history.
"I could not be more excited to join a club with Toronto FC's level of excellence and winning tradition and I will work tirelessly to uphold those standards," Armas said.
"As an opposing coach I felt the passion the fans and supporters bring to BMO Field. It is next level," he added. "I can't wait to put a team on the field that will not only make them proud with the way we play, but also with the way we run, battle and compete every minute of every game."
Armas was in charge of the New York Red Bulls for more than two years until his spell in the Big Apple ended in September.
New York dismissed him following the club's poor start to the season. Armas led the Red Bulls to two playoff appearances and ended his tenure with a 29-21-11 record.
Toronto had been without a coach since Vanney announced his departure in December. Vanney, who oversaw the team's MLS Cup-winning run in 2017, was appointed head coach of the Los Angeles Galaxy earlier in January.
TFC finished second in the Eastern Conference last season but were knocked out of the playoffs in the opening round after an upset loss to Nashville SC.