Batherson tied it 2-2 after tipping a shot from Shane Pinto past Matt Murray.
Tim Stutzle and Pinto scored in the shootout for Ottawa.
Stutzle had a goal and an assist for the Senators (4-1-0), who have won two straight. Linus Ullmark made 16 saves.
Jonathan Marchessault and Ross Colton scored for the Predators (2-2-1), who had a three-game point streak (2-0-1). Murray, playing his first NHL game since January 8, 2024, made 18 saves in his Nashville debut.
Colton gave the Predators a 1-0 lead at 8:50 of the first period. After a scramble in front of the net, Vitali Pinchuk found the puck and slid it to Colton, who went top shelf past Ullmark.
The assist gave Pinchuk his first career point in his NHL debut.
Marchessault extended it to 2-0 at 13:10 on the power play, beating Ullmark from close range.
With the assists, Steven Stamkos (three goals, four assists) and Roman Josi (one goal, five assists) both extended their season-opening point streaks to five games.
Stutzle’s power-play goal cut it to 2-1 at 9:29 of the second period. A feed from Dylan Cozens found Stutzle down low, and he one-timed it into an open net for his fourth goal in five games.
Ottawa went from 13:45 of the second to 16:06 of the third without a shot on goal.
