Smith turned aside 26-of-26 shots over the final two periods, recording his third road shutout of the post-season, as Phoenix beat the Kings for the first time following three straight losses to start the series.
Shane Doan scored both Coyotes goals -- a power-play marker in the first period and another tally midway through the second frame.
The Coyotes will again try to keep the series alive when it returns to Phoenix for Tuesday's Game 5.
Jonathan Quick stopped 19-of-21 shots for Los Angeles.
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