We report the identification from multiwavelength observations of the Fermi Large Area Telescope (LAT) source 4FGL J1405.1-6119 (=3FGL J1405.4-6119) as a high-mass gamma-ray binary. Observations with the LAT show that gamma-ray emission from the system is modulated at a period of 13.7135 ± 0.0019 days, with the presence of two maxima per orbit with different spectral properties. X-ray observations using the Neil Gehrels Swift Observatory X-ray Telescope show that X-ray emission is also modulated at this period, but with a single maximum that is closer to the secondary lower-energy gamma-ray maximum. A radio source, coincident with the X-ray source is also found from Australia Telescope Compact Array observations, and the radio emission is modulated on the gamma-ray period with similar phasing to the X-ray emission. A large degree of interstellar obscuration severely hampers optical observations, but a near-infrared counterpart is found. Near-infrared spectroscopy indicates an O6 III spectral classification. This is the third gamma-ray binary to be discovered with the Fermi-LAT from periodic modulation of the gamma-ray emission, the other two sources also have early O star, rather than Be star, counterparts. We consider at what distances we can detect such modulated gamma-ray emission with the LAT, and examine constraints on the gamma-ray binary population of the Milky Way.