The incident happened Wednesday inside a Eurocash store at a shopping center near the Sweden-Norway border, The Local.se reported.
Local media reported the two women began arguing while waiting in line to checkout. One of the women became so upset that allegedly she lit the other woman's hair on fire, the reports said.
"The woman who lit the other woman's hair on fire fled the scene before being arrested. But we have video of the entire incident from a surveillance camera," a spokesman for the Vastra Gotaland police department told local Norwegian newspaper Fredriksstad Blad.
Police said it was unclear what sparked the argument between the women.
The condition of the woman whose hair was set on fire was unknown, but local media reported that she was taken to a hospital.