Robust and accurate acceleration tracking remains a challenge in many fields. For geophysics and economic geology, precise gravity mapping requires onboard sensors combined with accurate positioning and navigation systems. Cold atom-based quantum inertial sensors can potentially provide these high-precision instruments. However, current scalar instruments require precise alignment with vector quantities. Here, we present the first hybrid three-axis accelerometer exploiting the quantum advantage to measure the full acceleration vector by combining three orthogonal atom interferometer measurements with a classical navigation-grade accelerometer triad. Its ultralow bias permits tracking the acceleration vector over long time scales, yielding a 50-fold improvement in stability (6 × 10-8 g) over our classical accelerometers. We record the acceleration vector at a high data rate (1 kHz), with absolute magnitude accuracy below 10 μg, and pointing accuracy of 4 μrad. This paves the way toward future strapdown applications with quantum sensors and highlights their potential as future inertial navigation units.