We present CO(1-0) and CO(2-1) maps of the starburst/Seyfert 1 galaxy NGC 2782 obtained with the IRAM interferometer, at 2.1 arcsec×1.5 arcsec and 0.7 arcsec×0.6 arcsec resolution respectively. The CO emission is aligned along the stellar nuclear bar of radius ~1 kpc, configured in an elongated structure with two spiral arms at high pitch angle ~90°. At the extremity of the nuclear bar, the CO changes direction to trace two more extended spiral features at a lower pitch angle. These are the beginning of two straight dust lanes, which are aligned parallel to an oval distortion, reminiscent of a primary bar, almost perpendicular to the nuclear one. The two embedded bars appear in Spitzer IRAC near-infrared images, and HST color images, although highly obscured by dust in the latter. We compute the torques exerted by the stellar bars on the gas, and find systematically negative average torques down to the resolution limit of the images, providing evidence of gas inflow tantalizingly close to the nucleus of NGC 2782. We propose a dynamical scenario based on numerical simulations to interpret coherently the radio, optical, and molecular gas features in the center of the galaxy. Star formation is occurring in a partial ring at ~1.3 kpc radius corresponding to the Inner Lindblad Resonance (ILR) of the primary bar; this ring-like structure encircles the nuclear bar, and is studded with Halpha emission. The gas traced by CO emission is driven inward by the gravity torques of the decoupled nuclear bar, since most of it is inside its corotation. N-body simulations, including gas dissipation, predict the secondary bar decoupling, the formation of the elongated ring at the ~1 kpc-radius ILR of the primary bar, and the gas inflow to the ILR of the nuclear bar at a radius of ~200-300 pc. The presence of molecular gas inside the ILR of the primary bar, transported by a second nuclear bar, is a potential ``smoking gun''; the gas there is certainly fueling the central starburst, and in a second step could fuel directly the AGN.