Context. Through spectrally unresolved observations of high-J CO transitions, Herschel Photodetector Array Camera and Spectrometer (PACS) has revealed large reservoirs of warm (300 K) and hot (700 K) molecular gas around low-mass protostars. The excitation and physical origin of this gas is still not understood. Aims. We aim to shed light on the excitation and origin of the CO ladder observed toward protostars, and on the water abundance in different physical components within protostellar systems using spectrally resolved Herschel-HIFI data. Methods. Observations are presented of the highly excited CO line J = 16-15 (E_up/k_B = 750 K) with the Herschel Heterodyne Instrument for the Far Infrared (HIFI) toward a sample of 24 low-mass protostellar objects. The sources were selected from the Herschel "Water in Star-forming regions with Herschel" (WISH) and "Dust, Ice, and Gas in Time" (DIGIT) key programs. Results. The spectrally resolved line profiles typically show two distinct velocity components: a broad Gaussian component with an average FWHM of 20 km s^-1 containing the bulk of the flux, and a narrower Gaussian component with a FWHM of 5 km s^-1 that is often offset from the source velocity. Some sources show other velocity components such as extremely-high-velocity features or "bullets". All these velocity components were first detected in H₂O line profiles. The average rotational temperature over the entire profile, as measured from comparison between CO J = 16-15 and 10-9 emission, is ~300 K. A radiative-transfer analysis shows that the average H₂O/CO column-density ratio is ~0.02, suggesting a total H₂O abundance of ~2 × 10^-6, independent of velocity. Conclusions. Two distinct velocity profiles observed in the HIFI line profiles suggest that the high-J CO ladder observed with PACS consists of two excitation components. The warm PACS component (300 K) is associated with the broad HIFI component, and the hot PACS component (700 K) is associated with the offset HIFI component. The former originates in either outflow cavity shocks or the disk wind, and the latter in irradiated shocks. The low water abundance can be explained by photodissociation. The ubiquity of the warm and hot CO components suggest that fundamental mechanisms govern the excitation of these components; we hypothesize that the warm component arises when H₂ stops being the dominant coolant. In this scenario, the hot component arises in cooling molecular H₂-poor gas just prior to the onset of H₂ formation. High spectral resolution observations of highly excited CO transitions uniquely shed light on the origin of warm and hot gas in low-mass protostellar objects.