Electric-field oscillations are now experimentally accessible in the THz-to-PHz frequency range1–11. Their measurement delivers the most comprehensive information content attainable by optical spectroscopy – if performed with high sensitivity. Yet, the trade-off between bandwidth and efficiency associated with the nonlinear mixing necessary for field sampling has so far strongly restricted sensitivity in applications such as field-resolved spectroscopy of molecular vibrations12,13. Here, we demonstrate electric-field sampling of octave-spanning mid-infrared waves in the 18-to-39 THz (600-to-1300 cm-1) spectral region, with amplitudes ranging from the MV/cm level down to a few mV/cm. Employing powerful short-wave mid-infrared gate pulses14,15, the field-measurement sensitivity approaches within a factor of 4 the ultimate detection limit of capturing all photons in the temporal gate. This combination of detection sensitivity and dynamic range enables optimum use of newly-emerging high-power waveform-controlled infrared sources12,14,16–22 for molecular spectroscopy. In a proof-of concept experiment, we performed broadband quantitative linear spectroscopy of multiple gases over more than 8 orders of magnitude in concentration, at an interaction length of only 45 cm. Our technique brings fast, label-free, quantitative multivariate detection of volatile organic compounds over the entire known physiologically-relevant molecular landscape23,24 within reach.