Boedker took advantage of a turnover and fought through two defenders to beat Corey Crawford 2:15 into the first extra session.
He scored 13:15 into overtime Tuesday to give the Coyotes a 3-2 win in the third game of the opening-round series.
With a 3-1 advantage, Phoenix will look to advance to the Western Conference semifinals when the Coyotes host Game 5 Saturday.
All of the scoring in regulation came in the third period.
Brendan Morrison then scored on a slap shot with 9:35 to go and Michael Frolik tied it on a backhander with 1:26 left.
Mike Smith, who had the league's fourth best save percentage during the regular season, stopped 30-of-32 shots to get the win.
Crawford made just 16 saves.