We present the first joint analysis of the power spectrum and bispectrum using the Data Release 1 (DR1) of the Dark Energy Spectroscopic Instrument (DESI), focusing on Luminous Red Galaxies (LRGs) and quasars (QSOs) across a redshift range of 0.4 ≤ z ≤ 2.1. By combining the two- and three-point statistics, we are able to partially break the degeneracy between the logarithmic growth rate, f(z), and the amplitude of dark matter fluctuations, σ s8(z), which cannot be measured separately in analyses that only involve the power spectrum. In comparison with the (fiducial) Planck ΛCDM cosmology we obtain f/f fid = {0.888-0.089 +0.186,0.977-0.220 +0.182,1.030-0.085 +0.368}, σ s8/σ fid s8 = {1.224-0.133 +0.091,1.071-0.163 +0.278,1.00 0-0.223 +0.088} respectively for the three LRG redshift bins, corresponding to a cumulative 10.1% constraint on f, and of 8.4% on σ s8, including the systematic error budget. Additionally, we obtain constraints for the ShapeFit compressed parameters describing the isotropic scaling parameter, α iso(z), the Alcock-Paczyński parameter, α AP(z), the combined growth of structure parameter fσ s8(z), and the combined shape parameter m(z)+n(z). Their cumulative constraints from our joint power spectrum-bispectrum analysis are respectively σ α iso = 0.9% (9% improvement with respect to our power spectrum-only analysis); σ α AP = 2.3% (no improvement with respect to power spectrum-only analysis, which is expected given that the bispectrum monopole has no significant anisotropic signal); σ fσ s8 = 5.1% (9% improvement); σ m+n = 2.3% (11% improvement). These results are fully consistent with the main DESI power spectrum analysis, demonstrating the robustness of the DESI cosmological constraints, and compatible with Planck ΛCDM cosmology.