We report strong terahertz (~10^12 Hz) high harmonic generation in thin films of Cd3As2, a three-dimensional Dirac semimetal at room temperature. The third harmonics is detectable with tabletop light source and can be as strong as 100 V/cm by applying the fundamental field of 6.5 kV/cm inside the film, showing an unprecedented efficiency for terahertz frequency conversion. Our time-resolved terahertz spectroscopy and calculations also clarify the microscopic mechanism of the nonlinearity originating in the coherent acceleration of Dirac electrons in momentum space. Our results provide clear insights for nonlinear current of Dirac electrons driven by terahertz field under an influence of scattering, paving the way toward novel devices for high-speed electronics and photonics based on topological semimetals.