Travis Konecny had one goal and one assist, Cam York had two assists and Dan Vladar made 23 saves for the Flyers (8-5-1), who have won consecutive games after losing their previous two.
Cates gave the Flyers a 2-1 lead at 17:37 of the second period on a rebound of a Jamie Drysdale shot.
Ryan O’Reilly scored for the Predators (5-7-4), who have lost three consecutive games (0-1-2). Juuse Saros made 23 saves.
O’Reilly gave the Predators a 1-0 lead at 1:44 of the first period on a deflection of a wrist shot from the slot by Filip Forsberg that first bounced off of Drysdale.
Matvei Michkov scored for the Flyers to make it 1-1 at 4:43 of the second period on a one-timer from the slot on a pass from York.
Konecny added an empty-net goal at 18:47 of the third period to make it 3-1.