The victims are members of the LeBaron family, linked to a breakaway Mormon community that settled in Mexico several decades ago.
The victims were travelling in a convoy of vehicles.
The security minister said the group could have been targeted accidentally as a result of mistaken identity.
Sonora state in northern Mexico is being fought over by two rival gangs, La Línea, which has links to the larger Juárez cartel, and "Los Chapos", which is part of the Sinaloa cartel.
In a tweet President Donald Trump described the victims as a group of "wonderful family and friends" who "got caught between two vicious drug cartels, who were shooting at each other".
The US "stands ready", he said, to offer support to combat the problem of cartel violence and "do the job quickly and effectively". The FBI has offered to assist Mexican authorities in the investigation, CNN reports.
Mexico's President, Andrés Manuel López Obrador, said Mexico would act with "independence and sovereignty" in pursuing the criminals behind the attack.
A relative of the victims, Alex LeBaron, told CNN that the bodies of the deceased had been returned to their family ranch.