IGR J17591--2342, a transient accretion-powered millisecond X-ray pulsar, was discovered during its 2018 outburst. Here, we present a timing and spectral analysis of the source using {\it AstroSat} data of the same outburst. From the timing analysis, we obtain updated values of binary orbital parameters, which reveal an average pulsar spin frequency of 527.4256984(8) Hz. The pulse profiles can be fit well with four harmonically related sinusoidal components with fractional amplitudes of fundamental and second, third, and fourth harmonics as $\sim13$%, $\sim$6%, $\sim$0.9%, $\sim$0.2%, respectively. The energy-dependent study of pulse profiles in the range of $3-20$ keV shows that the fractional amplitude of both the fundamental and first overtone is consistent with being constant across the considered energy band. Besides, a decaying trend has been observed for both the fundamental and first overtone in the phase-delay versus energy relation resulting in soft X-ray (2.8-3.3 keV) phase lags of $\sim$0.05 and $\sim$0.13 with respect to $\leq 15$ keV photons, for the fundamental and first overtone, respectively. The combined spectra from the Large Area X-ray Proportional Counters and the Soft X-ray Telescope aboard {\it AstroSat} in the $1-18$ keV range can be fit well with an absorbed model consisting of a Comptonization, a blackbody and a Gaussian emission line component yielding as best-fit parameters a blackbody seed photon temperature $kT_{\rm bb}$$\sim 0.95 \pm 0.03$ keV, and an electron temperature $kT_{\rm e}$$\sim 1.54 \pm0.03$ keV. The spectral aspects suggest the scattering of photons from the accretion disc or the neutron star's surface.