Phil Kessel scored his 17th goal of the season for Toronto, and Matt Frattin also scored for the Leafs. James Reimer made 23 saves.
Clarkson scored the winner off the rush after taking a feed from Dainius Zubrus in the slot and beating Reimer through his pads.
Kovalchuk and Clarkson scored power-play goals in the first period to give New Jersey a quick 2-0 lead.
Kovalchuck wired a one-time shot off the post and in past Reimer at 5:03. Clarkson then took advantage of some suspect defensive zone coverage by the Maple Leafs to beat a helpless Reimer at 8:59.
Kessel got the Leafs back in it at 11:29 of the second. Joffrey Lupul threw a puck in front that bounced off Kessel's skate and past Brodeur.
Frattin tied it at 1:56 of the third on a great individual effort. After picking up the puck near the left-wing boards, Frattin held off Adam Larsson before beating Brodeur to his left.
The Leafs had a chance to win in late in regulation on the power play as New Jersey's Patrik Elias took an undisciplined slashing penalty with 2:16 left in regulation, but Toronto failed to capitalize as the Devils hung on to force overtime.
Notes: The Leafs announced forward Clarke MacArthur is day-to-day with an upper body injury. ... Attendance at Air Canada Centre was 19,514. ... New Jersey finished 2-for-4 on the power play, while Toronto wound up 0-for-2.