How the initial rupturing of the magma‐poor rifts started remains a controversial scientific issue due to the lack of definitive studies on the lithospheric structure and mantle flow beneath the lithosphere from these rift systems. We constrain the anisotropic structure of the mantle in the magma‐poor Malawi rift zone (MRZ) by observing quasi‐Love (QL) waves, which are abnormal waveforms with Rayleigh wave polarization characteristics formed by the scattering of Love waves through lateral gradients in anisotropic structures. Here, we observed clear QL waves at stations in the western MRZ, indicating that there are significant lateral gradients in anisotropy beneath the western MRZ. The stations that recorded QL waves are close to the eastern edge of an unexposed Niassa craton revealed by the latest high‐resolution tomographic model AF2019. Comparing the observations of QL waves with recently published shear‐wave splitting measurements in the MRZ, we infer that the cratonic lithosphere of the Niassa induced edge‐driven convection (EDC) of the asthenospheric mantle on its eastern boundary, resulting in lateral gradients in seismic anisotropy beneath the western MRZ. As the EDC is a small‐scale, relatively weak form of mantle convection that is easily overwhelmed by strong active upwelled asthenosphere, its presence indicates the lack of large‐scale mantle convection associated with asthenospheric upwelling beneath the MRZ. The magma‐poor MRZ could be an example of passive rifting due to the local asthenospheric mantle flow.