It was the fourth career shutout for McElhinney, who is playing for his fifth NHL club in six seasons.
Nathan Horton, making his Columbus debut after recovering from shoulder surgery, opened the scoring with 4:10 to go in the second period.
Horton signed a seven-year contract with the Blue Jackets last summer after spending six seasons with Florida and three with Boston.
The other Columbus goal came from R.J. Umberger on the power play midway through the third period.
Mike Smith made 31 saves in the loss.