MiniJPAS is a ∼1 deg2 imaging survey of the AEGIS field in 60 bands, performed to demonstrate the scientific potential of the upcoming Javalambre-Physics of the Accelerating Universe Astrophysical Survey (J-PAS). Full coverage of the 3800-9100 Å range with 54 narrow-band filters, in combination with 6 optical broad-band filters, allows for extremely accurate photometric redshifts (photo-z), which, applied over areas of thousands of square degrees, will enable new applications of the photo-z technique, such as measurement of baryonic acoustic oscillations. In this paper we describe the method we used to obtain the photo-z that is included in the publicly available miniJPAS catalogue, and characterise the photo-z performance. We built photo-spectra with 100 Å resolution based on forced-aperture photometry corrected for point spread function. Systematic offsets in the photometry were corrected by applying magnitude shifts obtained through iterative fitting with stellar population synthesis models. We computed photo-z with a customised version of LEPHARE, using a set of templates that is optimised for the J-PAS filter-set. We analysed the accuracy of miniJPAS photo-z and their dependence on multiple quantities using a subsample of 5266 galaxies with spectroscopic redshifts from SDSS and DEEP, which we find to be representative of the whole r < 23 miniJPAS sample. Formal 1σ uncertainties for the photo-z that are calculated with the Δχ2 method underestimate the actual redshift errors. The odds parameter has a stronger correlation with |Δz| and accurately reproduces the probability of a redshift outlier (|Δz| > 0.03), regardless of the magnitude, redshift, or spectral type of the sources. We show that the two main summary statistics characterising the photo-z accuracy for a population of galaxies (σNMAD and η) can be predicted by the distribution of odds in this population, and we use this to estimate the statistics for the whole miniJPAS sample. At r < 23, there are ∼17 500 galaxies per deg2 with valid photo-z estimates, ∼4200 of which are expected to have |Δz| < 0.003. The typical error is σNMAD = 0.013 with an outlier rate η = 0.39. The target photo-z accuracy σNMAD = 0.003 is achieved for odds > 0.82 with η = 0.05, at the cost of decreasing the density of selected galaxies to n ∼ 5200 deg−2 (∼2600 of which have |Δz| < 0.003).