Radio observations of some asymptotic giant branch (AGB) star envelopes show excess emission at frequencies below 100 GHz that cannot be explained by thermal dust emission (hereafter anomalous microwave emission (AME)). Moreover, AGB envelopes are a common place where gas molecules condense to form nanoparticles (e.g., polycyclic aromatic hydrocarbons) and large grains. In this paper, we study whether electric dipole emission from rapidly spinning nanoparticles can reproduce the AME observed toward AGB stars. To properly model the size distribution of nanoparticles in the AGB envelope, we take into account both the increase of nanoparticles due to rotational disruption of large grains spun up by radiative torques and the decrease of the smallest nanoparticles due to rotational disruption driven by stochastic gas collisions. We then perform detailed modeling of microwave emission from rapidly spinning nanoparticles from both C-rich and O-rich AGB envelopes using the grain-size distribution constrained by rotational disruption. We find that spinning dust emission is dominant over thermal dust emission at frequencies below 100 GHz. We attempt to fit the observational data of AME using our spinning dust model and demonstrate that spinning dust can reproduce the observed AME in six AGB stars. Finally, we discuss how microwave emission from spinning dust in AGB envelopes could be observed with high-resolution upcoming radio telescopes such the Next Generation Very Large Array and Atacama Large Millimeter/submillimeter Array Band 1. This would be a major leap for understanding AGB envelopes' formation, evolution, and internal structures of dust. Observations would help to distinguish the carrier of AME via comparisons of C-rich and O-rich stars, because polycyclic aromatic hydrocarbons (PAHs) are formed in C-rich AGB stars, while silicates are formed in O-rich stars.