The Sun's corona is millions of degrees hotter than its 5,000 K photosphere. This heating enigma is typically addressed by invoking the deposition at coronal heights of non-thermal energy generated by the interplay between convection and magnetic field near the photosphere. However, it remains unclear how and where coronal heating occurs and how the corona is filled with hot plasma. We show that energy deposition at coronal heights cannot be the only source of coronal heating, by revealing a significant coronal mass supply mechanism that is driven from below, in the chromosphere. We quantify the asymmetry of spectral lines observed with Hinode and SOHO and identify faint but ubiquitous upflows with velocities that are similar (50-100 km/s) across a wide range of magnetic field configurations and for temperatures from 100,000 to several million degrees. These upflows are spatio-temporally correlated with and have similar upward velocities as recently discovered, cool (10,000 K) chromospheric jets or (type II) spicules. We find these upflows to be pervasive and universal. Order of magnitude estimates constrained by conservation of mass and observed emission measures indicate that the mass supplied by these spicules can play a significant role in supplying the corona with hot plasma. The properties of these events are incompatible with coronal loop models that only include nanoflares at coronal heights. Our results suggest that a significant part of the heating and energizing of the corona occurs at chromospheric heights, in association with chromospheric jets.