Berghenheim's score 5:54 into the second period put the Panthers ahead 2-0. Eleven minutes later, he set up Boyes for a three-goal advantage.
Tomas Fleischmann also scored and Aleksander Barkov had a pair of assists as Florida stayed hot on the road.
Scott Clemmensen made 28 saves.
Mason Raymond scored for the slumping Leafs, who fell for the fourth time in five contests.
James Reimer stopped 20-of-23 shots for Toronto.