Trailing 3-2 entering the final frame, Grabovski got the equalizer 4 minutes into the period and Kessel put the Leafs ahead only 2 1/2 minutes later when his shot went in off the foot of the Sabres' Steve Montador. Both skaters also notched assists.
James Reimer turned aside 39-of-42 shots for Toronto, which pulled to within four points of the eighth and final Eastern Conference playoff spot.
Clarke MacArthur and Tyler Bozak also scored for the Leafs.
Mike Weber, Jochen Hecht and Jordan Leopold found the net for Buffalo, which clings to the seventh place in the East.
Ryan Miller made 25 saves for the Sabres.