Rinne has 24 shutouts in four full seasons as Nashville's goaltender. He is three away from equaling the best single-season total of his career.
The Predators continued a surge during which they have won nine of 11. Columbus failed in its bid to win a second straight game and remained stuck on 13 victories, fewest in the NHL.
Martin Erat opened the scoring with 8:52 remaining in the second period and Mike Fisher produced a power-play goal 2:03 later. Shea Weber had an empty-net, shorthanded goal with 32 seconds left in the third.
Steve Mason made 22 saves for the Blue Jackets.