To investigate the relationship between the charge redistribution and ferromagnetism at the heterointerface between perovskite transition-metal oxides LaNiO$_3$ (LNO) and LaMnO$_3$ (LMO), we performed x-ray absorption spectroscopy and x-ray magnetic circular dichroism (XMCD) measurements. In the LNO/LMO heterostructures with asymmetric charge redistribution, the electrons donated from Mn to Ni ions are confined within one monolayer (ML) of LNO at the interface, whereas holes are distributed over 3-4 ML on the LMO side. A detailed analysis of the Ni-$L_{2,3}$ and Mn-$L_{2,3}$ XMCD spectra reveals that Ni magnetization is induced only by the Ni$^{2+}$ ions in the 1 ML LNO adjacent to the interface, while the magnetization of Mn ions is increased in the 3-4 ML LMO of the interfacial region. The characteristic length scale of the emergent (increased) interfacial ferromagnetism of the LNO (LMO) layers is in good agreement with that of the charge distribution across the interface, indicating a close relationship between the charge redistribution due to the interfacial charge transfer and the ferromagnetism of the LNO/LMO interface. Furthermore, the XMCD spectra clearly demonstrate that the vectors of induced magnetization of both ions are aligned ferromagnetically, suggesting that the delicate balance between the exchange interactions occurring inside each layer and across the interface may induce the canted ferromagnetism of Ni$^{2+}$ ions, resulting in weak magnetization in the 1 ML LNO adjacent to the interface.