Colin Wilson, Denis Grebeshkov and Shea Weber also scored for Nashville. Grebeshkov also had an assist while making his Predators debut against his former team, one day after being acquired from the Oilers in exchange for a second-round pick in this June's NHL draft.
Mike Comrie, Gilbert Brule, and Marc Pouliot scored for Edmonton.
Neither team led by more than a goal, and Pouliot tied the score 3-3 at 2:54 of the third before Arnott put the Predators ahead for good.
Brule tied the game at 2 at 11:53 of the second when he deflected Theo Peckham's wrist shot from the left point past Predators goalie Pekka Rinne on the stick side.
Nashville regained the lead at 15:12 on Weber's power-play goal off of an assist from Ryan Suter.
Two days earlier, the Predators' top defensive pairing had squared off in the Olympic gold medal game in Vancouver. Suter, who won silver with the United States, sent a pass from the top of the right circle to Weber in the center of the blue line where the gold medal-winning Canadian's one-timer beat Jeff Deslauriers on the glove side.
Comrie scored the game's first goal at 6:37 of the opening period. Tom Gilbert carried the puck into the Nashville zone with Comrie and Shawn Horcoff on a 3-on-1 against Suter. From the right side, Gilbert sent a pass to Horcoff, whose shot was stopped by Rinne. However, Comrie was parked just to the left of Rinne and had an easy tap-in for his seventh goal of the season.
Wilson drew the Predators even at 12:14 of the first on a wrist shot from the high slot that Deslauriers never saw thanks to a screen by Jordin Tootoo. Grebeshkov created an Edmonton turnover inside the blue line and was credited with an assist on Wilson's goal.
Grebeshkov scored his goal at 14:55 of the first when his slap shot from between the circles beat Deslauriers on the glove side.
NOTES: The Predators played their first home game since their building's name was changed to Bridgestone Arena late last month. ... Nashville has scored at least one power-play goal in a season-high seven consecutive games. ... The Predators are 19-1-0 when leading after the first period this season. ... Peckham's assist was his first career NHL point.