Zajac's tally held up as the game-winner when Andrew Shaw's last-chance try for the Blackhawks banged off the post.
Petr Sykora scored in regulation for the Devils, who used the win to climb four points over idle Ottawa for the sixth seed in the Eastern Conference.
Martin Brodeur earned the victory with 37 saves.
Brent Seabrook scored and Crawford finished with 21 saves for Chicago, which holds the sixth seed in the Western Conference.
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