Rapid specific sensors, ready for several particles’ detection per probe, are essential for disease diagnostics at the early stages before the symptom manifestation. Infectious diseases represent a prominent example where the early diagnosis is necessary both for a proper treatment of the patient as well as preventing the infection spreading. Here, a composite track-etched membrane with a 3D surface was used to create a biosensor specific to influenza A virus. The membrane allows concentration of the analyte on the surface, whereas the spike-like 3D surface covered with Ag layer serves as a substrate for surface-enhanced Raman spectroscopy (SERS) enhancing Raman scattering by 106-fold. The SERS-active coating is stable even in biological medium, being resistant to salts and proteins. A DNA aptamer with broad specificity to various influenza A strains was used as a recognition element, which was attached to the metal surface. As a result, the sensor can determine both human and avian influenza A viruses, covering all risky viral strains. The limit of detection (LoD) was as low as 120–2000 viral particles in mL. The LoDs are comparable with the LoDs of polymerase chain reaction with reverse transcription when the time of analysis is about 10 min. The biosensor allows rapid, specific and accurate detection of influenza A viruses in the nasal swabs.