In this work (Paper I), we analyse Herschel-PACS spectroscopy for a subsample of 23 O-rich and 3 S-type evolved stars, in different evolutionary stages from the asymptotic giant branch (AGB) to the planetary nebula (PN) phase, from the THROES catalogue. (C-rich targets are separately studied in Paper II). The broad spectral range covered by PACS (similar to 55-210 mu m) includes a large number of high-J CO lines, from J = 14-13 to J = 45-44 (v = 0), that allow us to study the warm inner layers of the circumstellar envelopes (CSEs) of these objects, at typical distances from the star of approximate to 10(14)-10(15) cm and approximate to 10(16) cm for AGBs and post-AGB-PNe, respectively. We have generated CO rotational diagrams for each object to derive the rotational temperature, total mass within the CO-emitting region and average mass-loss rate during the ejection of these layers. We present first order estimations of these basic physical parameters using a large number of high-J CO rotational lines, with upper-level energies from E-up similar to 580 to 5000 K, for a relatively big set of evolved low-to-intermediate mass stars in different AGB-to-PN evolutionary stages. We derive rotational temperatures ranging from T-rot similar to 200 to 700 K, with typical values around 500K for AGBs and systematically lower, similar to 200 K, for objects in more advanced evolutionary stages (post-AGBs and PNe). Our values of Trot are one order or magnitude higher than the temperatures of the outer CSE layers derived from low-J CO line studies. The total mass of the inner CSE regions where the PACS CO lines arise is found to range from M-H2 similar to 10(-6) to approximate to 10(-2) M-circle dot, which is expected to represent a small fraction of the total CSE mass. The mass-loss rates estimated are in the range M similar to 10(-7)-10(-4) M-circle dot yr(-1), in agreement (within uncertainties) with values found in the literature. We find a clear anticorrelation between M-H2 and. M vs. Trot that probably results from a combination of most e ffi cient line cooling and higher line opacities in high mass-loss rate objects. For some strong CO emitters in our sample, a double temperature (hot and warm) component is inferred. The temperatures of the warm and hot components are similar to 400-500K and similar to 600-900 K, respectively. The mass of the warm component (similar to 10(-5)-8 x 10(-2) M-circle dot) is always larger than that of the hot component, by a factor of between two and ten. The warm-to-hot MH2 and Trot ratios in our sample are correlated and are consistent with an average temperature radial profile of proportional to r(-0.5 perpendicular to 0.1), that is, slightly shallower than in the outer envelope layers, in agreement with recent studies.