Combining first-principles density-functional calculations and Moriya's self-consistent renormalization theory, we explain the recently reported counterintuitive appearance of an ordered magnetic state in uniaxially strained Sr_{2}RuO_{4} beyond the Lifshitz transition. We show that strain weakens the quantum spin fluctuations, which destroy the static order, more strongly than the tendency to magnetism. A different rate of decrease of the spin fluctuations vs magnetic stabilization energy promotes the onset of a static magnetic order beyond a critical strain.