In return, the Red Bulls were given a conditional draft pick from Toronto.
Amarikwa, who was claimed by New York off waivers this season from Colorado, has three goals in 31 MLS appearances.
The 24-year-old was also a member of the Rapids' championship squad in 2010.
Toronto head coach Paul Mariners said the move was made to address the injury to leading scorer Danny Koevermans, who suffered a season-ending knee injury last week.
"Quincy is a hard worker, with great athletic abilities and a positive attitude," Mariner said in statement. "That's what we look for in all our players. We feel he will fit in with our club nicely."
The trade comes one day after the Reds acquired French striker Eric Hassli in a trade from Vancouver.
The five-foot-nine, 180-pound Amarikwa was selected in the third round of the 2009 SuperDraft by San Jose.