In order to elucidate the origin of the interplay between charge density wave (CDW) and superconductivity in 1T-TaS2, we have synthesized powder samples of pure and C-intercalated 1T-CxTaS2 by means of a multi-anvil high-pressure synthesis method. We have found that single-phase samples are obtained in the 2-6 GPa range at 400 °C and for x=0-0.3. The structural, magnetic and transport properties of all samples have been investigated by means of neutron and x-ray diffraction, dc magnetization and dc electrical resistivity. For all x values including x=0, the data show that the CDW phase is suppressed concomitant to an abrupt onset of superconductivity, with Tc=3.2 K for x=0. The Tc value turns out to be weakly dependent on x, with a maximum Tc=3.8 K for x=0.2. This onset is accompanied by a crossover of magnetic behavior from paramagnetic Pauli-like to paramagnetic Curie-Weiss-like with effective moment 1.2 B/Ta, which suggests that a ionic picture is suitable for the superconducting phase, but not for the CDW phase. The analysis of the dependence of the a and c lattice parameters upon x as well as upon the synthesis conditions shows that the onset of superconductivity is mainly ascribed to unusual changes of the unit cell induced by the high-pressure synthesis. Specifically, the ex-situ lattice parameters exhibit a significantly larger c-axis parameter and a shrinking of the a-axis parameter stabilized by the high-pressure synthesis route. We argue that the above suppression of the CDW phase is induced by a broadening of the relevant 5d(t2g) band which stabilizes the metallic and superconducting phases. This scenario suggests that the strength of the electronic correlations are the main control parameter of the CDW-superconductivity competition in 1T-TaS2.