The win extended Washington's lead over Pittsburgh for fourth place in the Metropolitan Division to three points.
Trevor van Riemsdyk scored twice, Garnet Hathaway added a goal and Kuznetsov also contributed an assist. Charlie Lindgren stopped 31 shots.
Andrew Peeke, Eric Robinson and Johnny Gaudreau scored for Columbus, which has lost three straight and seven of its last nine to remain in last place in the NHL. Joonas Korpisalo stopped 30 shots.
Hathaway gave Washington the lead 5:06 into the first period. Van Riemsdyk made it 2-0 with 5:32 remaining before Peeke cut the lead in half 12 seconds later.
The teams traded goals in the second, with van Riemsdyk scoring his second at 8:26 and Robinson getting Columbus within one 2 ½ minutes later.
Gaudreau's goal at 12:44 of the third knotted the score 3-3.
Capitals: After the All-Star break, visit Boston on Saturday, Feb. 11.
Blue Jackets: Visit Toronto on Friday, Feb. 10.
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