The structure of the interfaces and the mechanisms of induced spin polarization of 1D infinite and finite narrow graphene- and h-BN zigzag nanoribbons placed on a SrO-terminated La1-xSrxMnO3 (LSMO) (001) surface were studied using density functional theory (DFT) electronic structure calculations. It was found that the pi-conjugated nanofragments are bonded to the LSMO(001) surface by weak disperse interactions. The types of coordination of the fragments, the strength of bonding, and the rate of spin polarization depend upon the nature of the fragments. Infinite and finite graphene narrow zigzag nanoribbons are characterized by the lift of the spin degeneracy and strong spin polarization caused by interface-induced structural asymmetry and oxygen-mediated indirect exchange interactions with Mn ions of LSMO support. Spin polarization changes the semiconducting nature of infinite graphene nanoribbons to half-metallic state with visible spin-up density of states at the Fermi level. The h-BN nanoribbon binding energy is weaker than graphene nanoribbon ones with noticeably shorter interlayer distance. The asymmetry effect and indirect exchange interactions cause spin polarization of h-BN nanoribbon as well with formation of embedded states inside the band gap. The results show a possibility to use one-atom thick nanofragments to design LSMO-based heterostructures for spintronic nanodevices with h-BN as an inert spacer to develop different potential barriers.