With Kaapo Kahkonen pulled for the extra attacker, Zuccarello scored from above the right face-off circle to make it 2-2.
Kahkonen made 29 saves, and Kevin Fiala and Frederick Gaudreau scored in the shootout for Minnesota (21-10-2), which won its second straight despite being without nine players.
"I think everyone in there did a [heck] of a job," Zuccarello said. "… I think whoever comes in to play steps up and does the job, and tonight was a good example of that. Like [Fiala] said before me, we're all leaders. Everyone can be a leader in different ways.
"It's just a good win against a really good team."
Zach Fucale stopped 21 of 22 shots for Washington and set the record for the longest streak without allowing a goal to begin an NHL career at 138:07.
"That's incredible. I did not know that," said Fucale, who made his second NHL start in his third game. "That's wild, wild stuff. It's good, quality night, but I'm mad at myself at that goal. To me, that's tough goal to give. Just another little experience I've got to learn from."