Brodeur evened his record at 9-9 this season and boosted his NHL all-time record victory total to 634.
Kovalchuk, Alexander Urbom and Adam Henrique all scored for the Devils in the second period and Brodeur had a shutout working into the third.
The Capitals then beat Brodeur three times in the final stanza, Brooks Laich scoring once and Jason Chimera twice. Chimera's second goal of the game and 13th of the season came with 1:42 left in regulation.
Michal Neuvirth made 25 saves for Washington, which has lost three of its last five.
New Jersey has an 8-1 record in shootouts this season including six victories in a row.
Washington has a 5-6 record since Dale Hunter took over as coach Nov. 28.