Optical rest-frame spectroscopic diagnostics are usually employed to distinguish between star formation and active galactic nucleus (AGN) powered emission. However, this method is biased against dusty sources, hampering a complete census of the AGN population across cosmic epochs. To mitigate th is effect, it is crucial to observe at longer wavelengths in the rest-frame near-infrared (near-IR), which is less affected by dust attenuation and can thus provide a better description of the intrinsic properties of galaxies. AGN diagnostics in this regime have not been fully exploited so far, due to the scarcity of near-IR observations of both AGNs and star-forming galaxies, especially at redshifts higher than 0.5. Using Cloudy photoionization models, we identified new AGN - star formation diagnostics based on the ratio of bright near-IR emission lines, namely [SIII] 9530{AA}, [CI] 9850{AA}, [PII] 1.188 micron, [FeII] 1.257 micron, and [FeII] 1.64 micron to Paschen lines (either Pa{gamma} or Pa{beta}), providing simple, analytical classification criteria. We applied these diagnostics to a sample of 64 star-forming galaxies and AGNs at 0<z<1, and 65 sources at 1<z<3 recently observed with JWST-NIRSpec in CEERS. We find that the classification inferred from the near-IR is broadly consistent with the optical one based on the BPT and the [SII]/H-alpha ratio. However, in the near-IR, we find ~60% more AGNs than in the optical (13 instead of eight), with five sources classified as "hidden" AGNs, showing a larger AGN contribution at longer wavelengths, possibly due to the presence of optically thick dust. The diagnostics we present provide a promising tool to find and characterize AGNs from z=0 to z~3 with low- and medium-resolution near-IR spectrographs in future surveys.