Smith was serving a penalty when Washington defenseman John Carlston tied the game with 3:27 to play, but he made up for that mistake by scoring his fourth goal that brought the Senators their 10th win of the season.
Bobby Ryan added an empty-net goal with 13 seconds to go.
Ottawa won in Washington for the first time in seven years, going 11-0-2 on the Capitals' home ice during that span.
The Senators overcame a 3-1 deficit after one period, getting second-period goals from Chris Phillips and Colin Greening to tie it.
Craig Anderson made 29 saves to get the victory. Braden Holtby gave up five goals on 40 shots as the Capitals suffered their fourth straight setback.
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