The Predators moved within one point of the Winnipeg Jets for the second wild card in the Western Conference.
"It's been a grind," said Nashville forward Colton Sissons, who tied it 3-3 at 18:57 of the second period when he redirected a shot from Forsberg. "Obviously it's been a challenging year for everyone, on personal notes and in general as a team. But we're right there in the hunt at this time of year, and honestly it doesn't get much better than that."
Forsberg scored in the first round of the shootout, and Ryan Ellis in the fourth before Grimaldi finally ended it in the eighth.
Pierre-Luc Dubois (first round) and Gustav Nyquist (fourth) scored in the shootout for Columbus.
"It's definitely frustrating," Blue Jackets forward Eric Robinson said. "We battled hard today to tie the game, and it was a hard-fought third period from both teams. Not to get the result in overtime or a shootout, yeah it stings a little bit, but take the point and move on to the next one."
Ryan Johansen gave Nashville a 1-0 lead 33 seconds into the first period on a rebound in front of the net, and Mikael Granlund made it 2-0 at 10:31 of the first.
"I don't think we're going to be looking back at this game in a month," Johansen said. "All we're going to look at is the points we got, and we'll take that and move forward."