The House voted 53-34 Tuesday to pass the proposal by Republican Rep. Thomas Willmott, of Kenner.
Drivers could secure dogs with a crate, leash or a ventilated bed cover, under the bill.
Willmott has proposed similar bills before, but they failed to garner enough support. He says he continues to push the issue to promote humane treatment of pets and public safety.
Willmott said four House members had stories of seeing dogs fall out of trucks. New Orleans Rep. Helena Moreno said one such incident forced her to swerve to keep from hitting a dog.