Context. Molecular counterparts to atomic jets have recently been detected within 1000 AU of young stars at early evolutionary stages. Reproducing these counterparts is an important new challenge for proposed ejection models. Aims. We explore whether molecules may survive in the magneto-hydrodynamic (MHD) disk wind solution currently invoked to reproduce the kinematics and tentative rotation signatures of atomic jets in T Tauri stars. Methods. The coupled ionization, chemical, and thermal evolution along dusty flow streamlines is computed for the prescribed MHD disk wind solution, using a method developed for magnetized shocks in the interstellar medium. Irradiation by (wind-attenuated) coronal X-rays and far-ultraviolet photons from accretion hot spots is included, with an approximate self-shielding of H 2 and CO. Disk accretion rates of 5 × 10 −6 , 10 −6 and 10 −7 M yr −1 are considered, representative of low-mass young protostars (so-called "Class 0"), evolved protostars ("Class I") and very active T Tauri stars ("Class II") respectively. Results. The disk wind has an "onion-like" thermo-chemical structure, with streamlines launched from larger radii having lower temperature and ionization, and higher H 2 abundance. The coupling between charged and neutral fluids is sufficient to eject molecules from the disk out to at least 9 AU. The launch radius beyond which most H 2 survives moves outward with evolutionary stage, from 0.2 AU (sublimation radius) in the Class 0 disk wind, to 1 AU in the Class I, and >1 AU in the Class II. In this molecular wind region, CO survives in the Class 0 but is significantly photodissociated in the Class I/II. Balance between ambipolar heating and molecular cooling establishes a moderate asymptotic temperature 700−3000 K, with cooler jets at earlier protostellar stages. As a result, endothermic formation of H 2 O is efficient, with abundances up to 10 −4 , while CH + and SH + can reach ≥10 −6 in the hotter and more ionised Class I/II winds. Conclusions. A centrifugal MHD disk wind launched from beyond 0.2−1 AU can produce molecular jets/winds up to speeds 100 km s −1 in young low-mass stars ranging from Class 0 to active Class II. The model predicts a high abundance ratio of H 2 to CO and an increase of molecular launch radius, temperature, and flow width as the source evolves, in promising agreement with current observed trends. Calculations of synthetic maps and line profiles in H 2 , CO and H 2 O will allow detailed tests of the model against observations.