With the 15 yr of Pass 8 data recorded by the Fermi Large Area Telescope, we report the detection of an extended gigaelectronvolt emission component with a 68% containment radius of 0.°85, which is spatially associated with the newly identified supernova remnant (SNR) G321.3–3.9. The γ-ray spectrum is best described by a log-parabola model in the energy range of 100 MeV–1 TeV, which shows a significant spectral curvature at ∼1 GeV. Either a leptonic or a hadronic model could explain the multiwavelength data of G321.3–3.9, while the leptonic model predicts a too-low strength of magnetic field. Also considering the flat radio spectrum of G321.3–3.9 and the γ-ray upper limit in the low energy band, the hadronic model is favored. The spatial coincidence between the γ-ray morphology and the diffuse thermal X-ray emission of G321.3–3.9 and the curved gigaelectronvolt γ-ray spectrum make G321.3–3.9 similar to the typical middle-aged SNRs interacting with molecular clouds (MCs). Such characteristics provide further evidence of the potential hadronic origin for its γ-ray emission. However, there is no MC detected around G321.3–3.9, which challenges the hadronic model.