Etienne made it 1-0 in the 20th minute, Nagbe scored a brilliant volley in the 81st and helped set up the third goal seven minutes later for Zardes. Andrew Tarbell made three saves for the Crew.
Columbus has the most points in Major League Soccer.
The Fire lost their third consecutive game.
Etienne opened the scoring with his first goal of the season against lax defending. Columbus defender Josh Williams started the play with a long ball out of the back. Milton Valenzuela gathered it and went unheeded to the left corner of the box, then sent a sharp pass back to Etienne, who ran onto the ball and was unchallenged by four Fire players.
Shuttleworth's 186th MLS regular season game was his first since Oct. 6, 2018, for Minnesota United.
Nagbe's first goal of the season came from a 30-yard strike that smacked the left post. He later went on the attack before finding Pedro Santos on the right flank. Santos sent a cross to Zardes for his fifth goal.
Zardes made amends for missing a penalty kick in the second minute of first-half stoppage time after Luis Diaz was fouled by Miguel Angel Navarro.
The Crew's first game in Columbus since March 1 was played with no fans because of COVID-19 restrictions.