The development of semiconductor gas sensors has potential applications in breath analysis to realize noninvasive health monitoring. Herein, tin oxide (SnO2) and copper oxide (CuO) heterocomposite nanofibers were fabricated by electrospinning Sn and Cu precursors on Si substrates, followed by calcination of sensing ammonia (NH3) with high sensitivity. The surface states of SnO2–CuO nanofibers, with different SnO2:CuO ratios, before and after calcination reveal different diffusion rates of SnO2 and CuO. Scanning electron microscopy images indicated that a large amount of CuO precipitated and formed nanograins that were adsorbed on the surface of SnO2 nanofibers with an increasing Cu ratio. Compared with pure SnO2 nanofibers, SnO2–CuO (2:1) heterocomposite nanofibers exhibited a significantly enhanced NH3-sensing response ( ${R}_{\text {a}}/{R}_{\text {g}}$ = 17.6). The mechanism was confirmed via the surface-state and band-structure analyses. The temperature-dependent response was also investigated. The SnO2–CuO (2:1) heterocomposite nanofibers show good response at room temperature. Thus, a demo of the application in breath analysis based on fabricated SnO2–CuO heterocomposite sensors was also exhibited.