The levels of several glial and neuronal plasma biomarkers have been found to increase during the acute phase in COVID-19 patients with neurological symptoms. However, replications in patients with minor or non-neurological symptoms are needed to understand their potential as indicators of CNS injury or vulnerability. Plasma levels of glial fibrillary acidic protein (GFAP), neurofilament light chain protein (NfL), and total Tau (T-tau) were determined by Single molecule array (Simoa) immunoassays in 45 samples from COVID-19 patients in the acute phase of infection [moderate (n = 35), or severe (n = 10)] with minor or non-neurological symptoms; in 26 samples from fully recovered patients after ~2 months of clinical follow-up [moderate (n = 23), or severe (n = 3)]; and in 14 non-infected controls. Plasma levels of the SARS-CoV-2 receptor, angiotensin-converting enzyme 2 (ACE2), were also determined by Western blot. Patients with COVID-19 without substantial neurological symptoms had significantly higher plasma concentrations of GFAP, a marker of astrocytic activation/injury, and of NfL and T-tau, markers of axonal damage and neuronal degeneration, compared with controls. All these biomarkers were correlated in COVID-19 patients at the acute phase. Plasma GFAP, NfL and T-tau levels were all normalized after recovery. Recovery was also observed in the return to normal values of the quotient between the ACE2 fragment and circulating full-length species, following the change noticed in the acute phase of infection. None of these biomarkers displayed differences in plasma samples at the acute phase or recovery when the COVID-19 subjects were sub-grouped according to occurrence of minor symptoms at re-evaluation 3 months after the acute episode (so called post-COVID or “long COVID”), such as asthenia, myalgia/arthralgia, anosmia/ageusia, vision impairment, headache or memory loss. Our study demonstrated altered plasma GFAP, NfL and T-tau levels in COVID-19 patients without substantial neurological manifestation at the acute phase of the disease, providing a suitable indication of CNS vulnerability; but these biomarkers fail to predict the occurrence of delayed minor neurological symptoms.