Harri Sateri made all 27 saves for Arizona after replacing Karel Vejmelka, who made two saves on six shots. Shayne Gostisbehere scored two goals and Travis Boyd, Jack McBain and Michael Carcone each scored once.
The Coyotes won their third straight after losing 10 straight (0-8-2), rallying much like they did in defeating the Dallas Stars 4-3 in overtime on Wednesday, when they trailed 3-0 in the third period.
"It was a mindset for us. We knew coming in this was an important game, and we didn't stay committed to what we needed to do for 60 minutes," Predators coach John Hynes said. "Now our mindset has to change. … We're in the postseason, and now we've got to get ourselves ready over the next few days for Tuesday."
The Predators would have dodged the Avalanche (56-19-7) by finishing off the Coyotes.
"They're an unreal team," Josi said. "They're one of the best teams in the League. A really fast team. Lots of skill. Every game we played them this year was a really tight, fast game."
Gostisbehere's shot from the point made it 4-1 at 13:21 of the first period and started the comeback. Boyd's redirection of Nick Schmaltz's shot cut it to 4-2 at 14:25 of the second.
"It all started when we scored the first goal … and then we really started to believe," Coyotes coach Andre Tourigny said.
McBain scored his second NHL goal at 18:24 to trim it to 4-3. Carcone tied it 4-4 at 1:51 of the third.
Gostisbehere's shot from the point through traffic gave Arizona a 5-4 lead at 10:27.
"They scored what, four goals in the first [seven] minutes?" Gostisbehere said. "Not the start you want, but it shows the team we have. Obviously we're not the best team, but it was a lot of fun winning."