Cam Atkinson and Seth Jones scored, and Joonas Korpisalo made 16 saves for the Blue Jackets (11-13-4), who have lost three straight and four of five.
"That's the disappointing part of this," Blue Jackets forward Nick Foligno said. "It isn't an effort thing. We've got to bury our opportunities, and I'm also tired of hearing about, 'We've got to score more.'
"There's a process to it. That's the end result. We've got to remain positive in the process, understanding how we're going to do that."
Panarin made it 3-1 at 15:32 of the second period with his 13th goal. He has 28 points (nine goals, 19 assists) in his past 20 games.
Brett Howden dug the puck out of the right corner to Tony DeAngelo, who passed to Panarin in the left circle for a one-timer while the Blue Jackets were on a line change.
"Two guys diving into the bench on Panarin's goal," Blue Jackets coach John Tortorella said. "It's just an awful change by [Alexander Wennberg] and [Riley Nash]."
Jones cut the lead to 3-2 at 3:33 of the third with his fourth goal. The Rangers challenged for goaltender interference, but the call was confirmed on video review.