Objective: This study examined systemic inflammatory indices and “Hemoglobin, Albumin, Lymphocyte, Platelet (HALP) scores” in neonates with hypoxic–ischemic encephalopathy (HIE). Methods: A total of 43 neonates with moderate-to-severe HIE at 36 weeks’ gestation were assessed. Systemic inflammatory markers were measured before HT commenced within 0–6 h after birth and between 60 and 72 h during and after therapy or before adjusting for hypothermia. Results: Platelet counts, hemoglobin levels, and platelet indices in the HIE group were significantly lower at both time points (p = 0.001). Both the neutrophil-to-lymphocyte ratio (NLR) and monocyte-to-lymphocyte ratio (MLR) decreased in the HIE group after hypothermia therapy (p = 0.001). Seizures, PVL, and kidney injuries were associated with higher HALP scores. The AUCs of NLR, PLR, MLR, SII, SIRI, and platelet, neutrophil, monocyte, and lymphocyte Index (PIV) showed significant sensitivity and specified HIE, with area under the curve (AUC) values of 0.654, 0.751, 0.766, 0.700, 0.722, and 0.749, respectively. Conclusions: A significant difference in systemic inflammatory markers was found between the HIE and control groups after hypothermia treatment, with significant reductions in the MLR and NLR. These markers, particularly MLR, were significant predictors of adverse clinical outcomes including seizures, PVL, and kidney damage.