EuTiO_{3} is an antiferromagnetic (AFM) material showing strong spin-lattice interactions, large magnetoelectric response, and quantum paraelectric behavior at low temperatures. Using electronic-structure calculations, we show that adding electrons to the conduction band leads to ferromagnetism. The transition from antiferromagnetism to ferromagnetism is predicted to occur at ∼0.08 electrons/Eu (∼1.4×10^{21} cm^{-3}). This effect is also predicted to occur in heterostructures such as LaAlO_{3}/EuTiO_{3}, where ferromagnetism is triggered by the formation of a high-density two-dimensional electron gas in the EuTiO_{3}. Our analysis indicates that the coupling between Ti 3d and Eu 5d plays a crucial role in lowering the Ti 3d conduction band in the ferromagnetic (FM) phase, leading to an almost linear dependence of the energy difference between the FM and AFM ordering on the carrier concentration. These findings open up possibilities in designing field-effect transistors using EuTiO_{3}-based heterointerfaces to probe fundamental interactions between highly localized spins and itinerant, polarized charge carriers.