The 33-year-old Scotland international will move to Canada with his family in July after a highly-successful two-and-a-half year spell on Teesside.
His Boro contract was due to expire at the end of the current campaign and Robson felt the opportunity to play abroad was too good for him and his family to pass up.
"I've enjoyed my time with Middlesbrough enormously and the move to England has been good for me and for my career," he said.
"My immediate priority is to help Boro win promotion to the Premier League, where the club and their supporters deserve to be.
"With the squad we have, there's every chance we can do that, although we all know it's going to be hard with the league being so tight.
"I'd like to wish Tony Mowbray, the club and the fans, who have welcomed me so warmly since I came here, the very best for the future.
"I'm very excited to be joining Whitecaps and to continue my career in Major League Soccer. My family and I are also looking forward to our move to the beautiful city of Vancouver.
"But I'll be keeping a very close eye on Boro's fortunes from Canada and they will always have a special place in my heart."
Robson has so far scored 18 goals in 77 league and cup games for Boro, including eight in 26 games this season.
His stunning goals against Hull City at the Riverside on Boxing Day and at Sunderland in the FA Cup will live long in the memories of Boro fans.
Born in Inverurie, Aberdeenshire, he began his career with Rangers but moved to Inverness Caledonian Thistle without making a first team appearance for the Ibrox club.
He joined Dundee United before being signed by Celtic, helping them win the 2008 Scottish Premier League title and gaining Champions League experience.
Former Boro manager Gordon Strachan brought him to Teesside for an undisclosed fee in January 2010.
Robson will join the Whitecaps in July subject to international clearance and satisfying visa requirement.
Boro would like to thank Robson for his considerable contribution on and off the field and wish him every success in the future.