Toye had a pair of brilliant strikes and Vito Mannone made eight saves for the Loons, who became just the second MLS team to win on LAFC's home pitch.
Toye scored in the 25th minute when he took a ball on the edge at midfield, dribbled into the middle and near the top of the box dropped a pass back to Jan Gregus. Gregus perfectly slotted the ball ahead to Toye, who left-footed it into the top far corner from a difficult angle just outside keeper box.
Toye got his sixth goal when he unleashed a high, hard shot from well outside the box into a top corner.
Mannone made two stops in the first half diving to his right. In the second half he made a leaping save on a free kick as LAFC put pressure on the visitors. LAFC had 24 shots, eight on target; the Loons (13-9-6) had just five shots.
Minnesota coach Adrian Heath earned his 50 MLS win as his team jumped from seventh to a three-way tie for fourth in the Western Conference.
LAFC (19-4-5) could have clinched the top seed in the West with a win. Its home record is now 11-1-2.