The solid electrolyte interphase (SEI), a nanoscale passivation film formed by reductive electrolyte decomposition during initial battery charging, is a key component of lithium-ion batteries (LIBs). When properly formed, an SEI layer allows ion mobility and prevents further degradation of the battery anode and electrolyte. Despite the essential protective role of the SEI and its importance in promoting reversible cycling in LIBs, the mechanisms underlying SEI formation remain poorly understood. In this work, we use first-principles quantum chemical calculations and stochastic methods to directly model mechanistic reactive competition during SEI formation for the first time. Our simulations, which are general to any LIB with an ethylene carbonate (EC)-based electrolyte, suggest that the separation of the SEI into primarily inorganic and primarily organic layers arises due to reductive decomposition of major organic products near the electrode interface. By varying the initial quantities of water and carbon dioxide, we also explore the role of impurity species in SEI formation. We validate that water is critical to the formation of lithium ethylene monocarbonate (LEMC) and could promote the formation of the postulated dilithium ethylene monocarbonate (DLEMC). Further, we observe that carbon dioxide reduction can compete with EC reduction, leading to downstream competition between organic carbonates, inorganic carbonates, and lithium oxalate. In addition to furnishing fundamental insights into the complex chemistry of the SEI, these findings provide a roadmap for future mechanistic studies of SEI formation in LIBs as well as next-generation batteries.