Radim Vrbata scored the only goal of the game for the Coyotes, who bounced back from a shootout loss to Vancouver on Monday to win for the sixth time in the past seven games.
"This was a whole team effort and we're finding a way to win," said Smith. "The last few minutes were frantic, but I thought we handled it pretty good."
Jonathan Quick gave up just the one goal on 21 shots for the Kings, who have dropped three of four.
"You have to score goals to win games and obviously we're not doing that right now," said Los Angeles forward Anze Kopitar. "It's nothing new here, we've been doing this all season long."
The only goal of the game came on the power play with 4:01 to play in the second as Ray Whitney sent a pass from the left circle to the right where Vrbata took a stride to the right and wristed it past the glove of Quick.
The Kings had a chance as Michal Rozsival was called for cross-checking with 90 seconds to play and Quick was pulled for a 6-on-4 advantage.
Phoenix, though, blocked the passing and shooting lanes and Dustin Brown was called for interference with 11 seconds to play to kill any chance of Los Angeles tying the game.
Smith made 11 saves in the first and eight in the second before stopping all nine in the third for his 15th career shutout.
Los Angeles hosts Calgary on Saturday...Phoenix hosts Dallas on Saturday...Los Angeles has won three of the five games in the season series with Phoenix. These two will meet one last time in the regular season next Tuesday in Phoenix.