The Vancouver Whitecaps picked up the Scotland international for free on a two-year deal from Southampton last September.
But after just 11 appearances in all competitions for the 'Caps, including four starts in the MLS Cup Playoffs, Armstrong has sealed a return to English football.
"Stuart was presented with an opportunity to return to England with the goal of playing again in the Premier League," Whitecaps sporting director Axel Schuster said in a club statement.
"We are thankful to Stuart, respected his decision, and wish him and his family all the best."
An 'extremely grateful' Armstrong paid tribute to the 'kindness' shown to him from all at the club and their supporters.
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"I am also so thankful to have been a part of one of the best groups of people I have ever met," Armstrong said.
"I will of course miss the beautiful city, but more importantly I will miss the people in it.
"I will continue to support the team from afar and wish them nothing but success for the future."
Championship side Sheffield Wednesday announced the capture of Armstrong, although they did not specify the length of his deal.
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The Owls stated Armstrong is 'bringing outstanding pedigree and knowhow to Hillsborough' and 'bolsters Danny Rohl's engine room options.'
The 32-year-old's arrival heralds his second stint in the Championship having played a leading role in helping Southampton seal promotion via the play-offs last season.
Armstrong made 42 appearances in the second tier last term, registering five goals and seven assists.
It was the Scot's final season at St Mary's having amassed just shy of 150 Premier League games across five seasons in the English top flight.
In that time, he scored 16 goals and provided 11 assists.
Armstrong, a central midfielder by trade, now faces the challenge of breaking into the Owls' starting lineup.
Club veteran Barry Bannan and Shea Charles, who is on loan from Armstrong's former employers Southampton, are Wednesday boss Rohl's preferred pairing in central midfield.
However, Armstrong has demonstrated throughout his career he is capable of playing on the left and right flanks, meaning Bannan and Charles could maintain their places.
Armstrong joins as Wednesday look to remain in the hunt for the final two play-off places.
The Owls sit 10th in the Championship on 41 points, three away from West Bromwich Albion and Middlesbrough who sit fifth and sixth respectively on 44 points.
Wednesday are next in action on Saturday as they host 23rd-placed Luton Town.