A new method for frequency selective surface (FSS) with high selectivity, namely filtenna-filtenna (FA-FA)-based technique, is proposed, which consists of two filtering antennas and impedance matching network. Based on the reciprocity theorem, the radiation null of filtenna can provide a stopband of the proposed FSS. In this work, it is validated by using a periodic array of back-to-back magnetoelectronic (ME)-dipole antennas with split-ring resonators (SRRs). And a slotted GND plane is used to connect the two ME-dipole antennas for impedance matching. Bandpass response of the proposed FSS is realized by the intrinsic high-pass characteristic of ME-dipole antennas and the bandstop characteristic of SRRs. Four transmission zeros (TZs) and three transmission poles (TPs) are generated, exhibiting a high-order filtering response. TZs and TPs are analyzed by equivalent circuit model and current distributions. The 3-dB fractional bandwidth (FBW_3db) of 38.5% in the passband and the 20-dB fractional bandwidths (FBW_20db) of 8.7% in the lower rejection band and 13.8% in upper rejection band are obtained in this design. Finally, a prototype is fabricated and measured to further validate the proposed FA-FA-based technique. Simulated and measured results show that the proposed FA-FA-based FSS is with the distinct advantages including wide passband, low profile, tunable TZs and dual-polarized application, which verifies the proposed method of FSSs.