Diamond sensors (DS) are widely used as solid-state particle detectors, beam loss monitors, and dosimeters in high-radiation environments, e.g., particle colliders. We have calibrated our DS with steady $\beta$- and X-radiation, spanning a dose rate in the range 0.1-100 mGy/s. Here, we report the first systematic characterization of transient responses of DS to collimated, sub-picosecond, 1 GeV electron bunches. These bunches, possessing a charge ranging from tens to hundreds of pC and a size from tens of microns to millimeters, are suitably provided by the FERMI electron linac in Trieste, Italy. The high density of charge carriers generated by ionization in the diamond bulk causes a transient modification of electrical properties of DS (e.g., resistance), which in turn affects the signal shape. We have modeled a two-step numerical approach, simulating the effects on the signal of both the evolution of charge carrier density in the diamond bulk and the changes in the circuit parameters. This approach interprets features observed in our experimental results to a great extent.