We present novel cosmological constraints based on a joint analysis of our H II galaxies (HIIG) Hubble relation with the full Planck cosmic microwave background (CMB) anisotropy spectrum and the baryon acoustic oscillations (BAO) probes. The HIIG span a large redshift range (0.088 ≤ z ≤ 2.5), reaching significantly higher redshifts than available Type Ia supernovae (SNeIa) and hence they probe the cosmic expansion at earlier times. Our independent constraints compare well with those based on the ‘Pantheon’ compilation of SNeIa data, which we also analyse. We find our results to be in agreement with the conformal Λ cold dark matter (ΛCDM) model within 1σ. We also use our HIIG data to examine the behaviour of the dark energy equation-of-state parameter under the Chevallier-Polarski-Linder (CPL) parametrization, w = w0 + waz/(1 + z), and find consistent results with those based on SNeIa, although the degeneracy in the parameter space and the individual parameter uncertainties, when marginalizing one over the other, are quite large.