Aleutian disease (AD) is a chronic disease of mink caused by the Aleutian Mink Disease Virus (AMDV) that results in dysfunction of the immune system. The prevalence of asymptomatic AMDV infections suggests a necessity to explore their effects on the cellular mechanisms of non-specific immunity in farmed mink. The study evaluated the phagocytic activity and oxygen metabolism of peripheral blood granulocytes and monocytes in mink with chronic subclinical AMDV infection. Moreover, the intensity of inflammatory processes was assessed based on the serum amyloid A (SAA) concentration. The analyses involved 24 brown mink females aged 12–24 months. The experimental group (group I) consisted of mink with chronic subclinical AMDV infections, and the control group (group II) included healthy animals. The statistical analysis was performed using the Mann-Whitney U rank test. Phagocytic activity of granulocytes and monocytes was carried out using flow cytometry, and SAA concentration was determined with enzyme-linked immunosorbent assay (ELISA). Compared with the control group, there was a significant decrease in the phagocytic activity and oxygen metabolism of granulocytes and monocytes in the AMDV-infected mink (p < 0.05). Additionally, it was found that the mean SAA value was significantly higher in the group infected with AMDV than in the control group (p < 0.05). The obtained data indicate that monitoring the serum SAA levels in mink with asymptomatic inflammation may help assess the health of mink and detect asymptomatic inflammation caused by AMDV infection.