In January 2020, we identified two severe acute respiratory syndrome coronavirus 2 (SARS‐CoV‐2)‐infected patients in a familial cluster with one person coming from Wuhan, China. The complete genome sequences of two SARS‐CoV‐2 strains isolated from these patients were identical and 99.98% similar to strains isolated in Wuhan. This is genetically suggestive of human‐to‐human transmission of SARS‐CoV‐2 and indicates Wuhan as the most plausible origin of the early outbreak in Vietnam. The younger patient had a mild upper respiratory illness and a brief viral shedding, whereas the elderly with multi‐morbidity had pneumonia, prolonged viral shedding, and residual lung damage. The evidence of nonsynonymous substitutions in the ORF1ab region of the viral sequence warrants further studies. Highlights Transmission of SARS‐CoV‐2 is a global public health and clinical concern. This report describes clinical features, virus isolation, and complete genome sequences from the first two SARS‐CoV‐2 infections in Vietnam. Epidemiological and phylogenetic analysis suggested evidence of human‐to‐human transmission of SARS‐CoV‐2. Comparison of SARS‐CoV‐2 strains isolated from these two patients with those from Wuhan showed high similarities. Nonsynonymous substitutions existed in the ORF1ab region of the viral sequence. Compared with mild clinical and virological manifestations in the younger patient, the elderly suffered from pneumonia, prolonged viral shedding, and residual lung damage.