The gun battles were quickly brought under control by prison guards and investigators found two pistols and an assault rifle inside the jail, Sinaloa state prosecutors' office said.
Police have been investigating what sparked the violence, the spokeswoman for the state's Public Safety Department said.
Many of Mexico's most powerful drug traffickers hail from the northwestern state of Sinaloa, a key smuggling corridor and cultivation area for marijuana and other narcotics.
Sinaloa, and Mazatlan in particular, is rife with turf battles among drug gangs.
Mexico's drug gangs frequently try to break their members out of prison by staging attacks.
But there has been no confirmation Monday's violence involved a prison-break attempt.