P.K. Subban, Viktor Arvidsson and Roman Josi scored for the Predators (7-1-0), who have won 13 straight games against the Oilers.
It was the fifth shutout of Saros' NHL career.
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"We knew before the game they had lots of talent, so we had to be solid in the defensive zone," Saros said. "I think we were that today and our penalty kill was really great too."
Goalie Cam Talbot made 25 saves for the Oilers (3-3-0), whose three-game winning streak ended.
"I thought we played a pretty hard, solid game, not enough offense, but we checked, we skated, we battled, we had a better start," Oilers coach Todd McLellan said. "The two-minute phase in the second period really hurt us. I didn't think our power play was quite as sharp either. We were the best penalty killers on the ice. In most situations, we killed a lot of our own plays until the last one where we had some chances."
Subban scored shorthanded at 13:14 of the second period when the Predators defenseman collected a loose puck near the Oilers blue line and went on a 2-on-1 rush with forward Ryan Hartman. Subban kept the puck and shot it past Talbot to give Nashville a 1-0 lead. It was the first shorthanded goal of Subban's NHL career.