The interstellar medium in the Milky Way's Central Molecular Zone (CMZ) is known to be strongly magnetised, but its large-scale morphology and impact on the gas dynamics are not well understood. We explore the impact and properties of magnetic fields in the CMZ using three-dimensional non-self gravitating magnetohydrodynamical simulations of gas flow in an external Milky Way barred potential. We find that: (1) The magnetic field is conveniently decomposed into a regular time-averaged component and an irregular turbulent component. The regular component aligns well with the velocity vectors of the gas everywhere, including within the bar lanes. (2) The field geometry transitions from parallel to the Galactic plane near z = 0 to poloidal away from the plane. (3) The magneto-rotational instability (MRI) causes an in-plane inflow of matter from the CMZ gas ring towards the central few parsecs of 0.01-0.1 M-circle dot yr(-1) that is absent in the unmagnetised simulations. However, the magnetic fields have no significant effect on the larger-scale bar-driven inflow that brings the gas from the Galactic disc into the CMZ. (4) A combination of bar inflow and MRI-driven turbulence can sustain a turbulent vertical velocity dispersion of sigma(z) = 5 km s(-1) on scales of 20 pc in the CMZ ring. The MRI alone sustains a velocity dispersion of sigma(z) similar or equal to 3 km s(-1). Both these numbers are lower than the observed velocity dispersion of gas in the CMZ, suggesting that other processes such as stellar feedback are necessary to explain the observations. (5) Dynamo action driven by differential rotation and the MRI amplifies the magnetic fields in the CMZ ring until they saturate at a value that scales with the average local density as B similar or equal to 102 (n/10(3) cm(-3))(0.33)mu G. Finally, we discuss the implications of our results within the observational context in the CMZ.