The Hector Galaxy Survey is a new optical integral field spectroscopy (IFS) survey currently using the Anglo-Australian Telescope to observe up to 15 000 galaxies at low redshift ($z \lt 0.1$). The Hector instrument employs 21 optical fibre bundles feeding into two double-beam spectrographs, AAOmega and the new Spector spectrograph, to enable wide-field multi-object IFS observations of galaxies. To efficiently process the survey data, we adopt the data reduction pipeline developed for the SAMI Galaxy Survey, with significant updates to accommodate Hector’s dual-spectrograph system. These enhancements address key differences in spectral resolution and other instrumental characteristics relative to SAMI and are specifically optimised for Hector’s unique configuration. We introduce a two-dimensional arc fitting approach that reduces the root-mean-square (RMS) velocity scatter by a factor of 1.2–3.4 compared to fitting arc lines independently for each fibre. The pipeline also incorporates detailed modelling of chromatic optical distortion in the wide-field corrector, to account for wavelength-dependent spatial shifts across the focal plane. We assess data quality through a series of validation tests, including wavelength solution accuracy (1.2–2.7 km s$^{-1}$ RMS), spectral resolution (FWHM of 1.2–1.4 Å for Spector), throughput characterisation, astrometric precision ($\lesssim$ 0.03 arcsec median offset), sky subtraction residuals (1–1.6% median continuum residual), and flux calibration stability (4% systematic offset when compared to Legacy Survey fluxes). We demonstrate that Hector delivers high-fidelity, science-ready datasets, supporting robust measurements of galaxy kinematics, stellar populations, and emission-line properties and provide examples. Additionally, we address systematic uncertainties identified during the data processing and propose future improvements to enhance the precision and reliability of upcoming data releases. This work establishes a robust data reduction framework for Hector, delivering high-quality data products that support a broad range of extragalactic studies.