Arturs Silovs made 29 saves for Vancouver (26-32-5), which is 5-2-1 in its past eight games.
"I didn't think we had our legs, but we hung in there," Canucks coach Rick Tocchet said. "Some guys didn't have their legs, but we grinded."
Luke Evangelista's first two NHL goals tied it in the third period, and Juuse Saros made 26 saves for Nashville (31-23-7), which lost for just the second time in seven games (5-1-1).
The Predators are five points behind the Colorado Avalanche for the second wild card into the Stanley Cup Playoffs from the Western Conference.
"We would have liked to get two [points], but in a tight playoff race, we'll take any point we can get," Evangelista said. "So, it was cool, the second one, to tie it up late, and it was a huge adrenaline rush. I was just happy to get the point."
Evangelista, who was robbed in tight by Silovs on a backdoor chance late in the first period, scored his first NHL goal in his fourth game on a rush rebound at 8:31 of the third to cut it to 3-2, then deflected Tyson Barrie's slap shot with 3:05 left to tie it 3-3.
"A lot of shock. Pretty big relief," Evangelista said. "I had a lot of chances and I just wasn't able to bury [them], so to finally see one go in the net was a big relief. It was pretty calming."