Johnson, brought up from the American Hockey League this week to sub for the injured Mike Smith, recorded the shutout in just his seventh career NHL game.
It was the 26-year-old's first win since notching a victory for the New York Rangers in 2010.
Keith Yandle helped Phoenix snap a two-game losing streak with a goal and two assists.
Pekka Rinne surrendered four goals on 25 shots as the Predators dropped a third straight game.
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