Two approaches to the challenging aim of forecasting impending eruptions are searching for empirical precursors and developing suitable interpretative models. Here we present high-resolution time series of 3He/4He ratios measured in gases emitted from peripheral vents around Mount Etna volcano (Italy), which revealed variations with strong correlations over both time and a broad spatial scale. The main eruptive episodes are preceded by increases in 3He/4He, making this ratio a unique tracer for monitoring volcanic activity. These features strongly reflect pressurization beneath the volcano due to deep magma influx. We propose a pioneering model that relates the changes in 3He/4He to the time-dependent outflow of volatiles from a magmatic chamber subjected to evolution of its internal pressure due to magma injection. At Mount Etna, the model makes it possible to estimate in near real time key parameters such as the rate of magma input and volume change in deep chamber preceding eruptions, and to compare them with geodetic estimations. This represents an unprecedented use of 3He/4He to obtain quantitative information on the physics of magmatic systems. Volcanoes showing changes of 3He/4He ratio in discharged gases due to unrest episodes are widespread in the world, and therefore we envisage extensive future applications of this approach.