In this letter we present the first measurements of an exclusive electron neutrino cross section with the MicroBooNE experiment using data from the Booster Neutrino Beamline at Fermilab. These measurements are made for a selection of charged-current electron neutrinos without final-state pions. Differential cross sections are extracted in energy and angle with respect to the beam for the electron and the leading proton. The differential cross section as a function of proton energy is measured using events with protons both above and below the visibility threshold. This is done by including a separate selection of electron neutrino events without reconstructed proton candidates in addition to those with proton candidates. Results are compared to the predictions from several modern generators, and we find the data agrees well with these models. The data shows best agreement, as quantified by $p$-value, with the generators that predict a lower overall cross section, such as GENIE v3 and NuWro.