We use near-infrared, ground-based data from the VISTA Variables in the Via Lactea (VVV) survey to indirectly extend the astrometry provided by the Gaia catalog to objects in heavily extinct regions toward the Galactic bulge and plane that are beyond Gaia's reach. We made use of state-of-the-art techniques developed for high-precision astrometry and photometry with the Hubble Space Telescope to process the VVV data. We employed empirical, spatially variable, effective point spread functions and local transformations to mitigate the effects of systematic errors, like residual geometric distortion and image motion, and to improve measurements in crowded fields and for faint stars. We also anchored our astrometry to the absolute reference frame of Gaia Data Release 3. We measure between 20 and 60 times more sources than Gaia in the region surrounding the Galactic center, obtaining a single-exposure precision of about 12 mas and a proper-motion precision of better than 1mas/yr for bright, unsaturated sources. Our astrometry provides an extension of Gaia into the Galactic center.We publicly release the astro-photometric catalogs of the two VVV fields considered in this work, which contain a total of ~3.5 million sources. Our catalogs cover ~3 sq. degrees, about 0.5% of the entire VVV survey area.