Matt Duchene scored, Filip Forsberg had a goal and an assist, and Juuse Saros made 44 saves for Nashville (27-14-4), which had won three in a row.
"I thought we played pretty well, we didn't give up too much," Duchene said. "I would love to see the chances; I feel like we had a few more than they did, their goalie played outstanding, he made some big saves.
"Juuse did as well for us, both goalies were good in the shootout. And when you get to the shootout, it's a bit of a coin flip, it's a bit of a skills comp, and our shooters, including myself, weren't able to find one. That's the way it goes."
Koskinen stopped Forsberg, Duchene and Roman Josi in the shootout. Saros made saves on Leon Draisaitl and Ryan Nugent-Hopkins.
"The last couple games we have played really well and I am happy to see that," Koskinen said. "I think we have learned our lessons. We can't look too far ahead, it is cliche, but you have to take it one game at a time, otherwise our focus is not there. We are building our game and doing a lot of good things right now. There is still a lot of room to improve and if we do that it is going to be a really strong team at the end of the year."
Forsberg gave the Predators a 1-0 lead at 5:07 of the first period. He scored on a shot through traffic from the right face-off circle.
McDavid tied it 1-1 at 19:28, getting past Predators defenseman Matt Benning on a 1-on-1 rush and shooting it past Saros.
"I think they shoot quite a bit, they're a high-shot generation team, so we did spend some time in our zone for sure," Predators coach John Hynes said. "I think they're the No. 1 team in offensive zone time in the League, but I thought we did a pretty good job. We were in there more than we wanted to be, but I thought we did a pretty good job of making sure we took care of some Grade-A areas and Juuse made some big saves, but I thought when we were called upon to defend in those areas, I thought we did a pretty good job."