Setting the managed switch port to untagged with a PVID of the desired VLAN will effectively extend that VLAN to all of the ports on the unmanaged switch. Your managed switch will "see" the multiple networks, and treat anything in and out of that port as part of that specific VLAN, and everything on the other side of that cable will only "see" a single normal layer 2 network.
Any VLAN tags will be ignored (and probably stripped) by the unmanaged switch.