Keith Yandle's tally with 1:45 left in 2, at Bridgestone Arena.
Jason LaBarbera picked up the victory with 30 stops.
Pekka Rinne yielded three goals on 35 shots in defeat.
Yandle notched the game-winner in a 4-on-4 situation, beating Rinne from the left wing after Shane Doan held onto the puck on the right wing until he found space to whip a cross-ice pass.
Rinne was sent to the bench in the final minute but Nashville couldn't equalize.
Phoenix managed to light the lamp after only nine seconds.
The visitors won the opening faceoff and Boyd Gordon dumped the puck into the Nashville zone. It slowed after hitting an official on the right wing, and Lauri Korpikoski followed up with a shot that Rinne appeared to have trapped with his glove.
But the disc dropped down and Pyatt was there to knock it home for a 1-0 game.
The Preds knotted the score with 10.8 seconds remaining as a Hillen shot caromed off LaBarbera's body and trickled over the line with Mike Fisher in pursuit.
Whitney stood alone in the high slot to one-time a pass from Radim Vrbata at 1:40 of the third period, but Nashville cashed in on a two-man advantage when Weber's left-point blast hit the back of the net on the short side with 7:07 to play.
Phoenix has won five of the last eight meetings with Nashville...Whitney has four goals in his last six games, and reached double digits for the 14th consecutive season...Prior to the game, the Coyotes assigned forward Patrick O'Sullivan to Portland of the AHL.