Hayes pushed the lead to 3-0 at 15:41 on a slap shot from below the right circle that went off Rinne short side.
"I thought the first period we played really well," Predators coach Peter Laviolette said. "I thought we had the edge and just with what we were trying to do. I thought we were sharp and on point and then the next 40 (minutes) turned, we weren't good at all."
Connor scored on a 2-on-1 at 15:22 of the third period to make it 4-0 and completed the hat trick on a one-timer to make it 5-0 with nine seconds remaining in regulation.
"I was part of that," Hellebuyck said. "I did my job when I was needed, but the guys in front of me made it pretty easy.
"They were phenomenal. They were getting their sticks on everything, they were blocking shots, they were controlling the pace of play. Our details were right."