Open‐conduit conditions characterize several of the most hazardous and active volcanic systems of basaltic composition worldwide, persistently refilled by magmatic inputs. Eruptive products with similar bulk compositions, chemically buffered by continual mafic inputs, nevertheless exhibit heterogeneous glass compositions in response to variable magma mixing, crystallization, and differentiation processes within different parts of the plumbing system. Here, we document how multivariate statistics and magma differentiation modeling based on a large data set of glass compositions can be combined to constrain magma differentiation and plumbing system dynamics. Major and trace elements of matrix glasses erupted at Stromboli volcano (Italy) over the last 20 years provide a benchmark against which to test our integrated petrological approach. Principal component analysis, K‐means cluster analysis, and kernel density estimation reveal that trace elements define a multivariate space whose eigenvectors are more readily interpretable in terms of petrological processes than major elements, leading to improved clustering solutions. Comparison between open‐ and closed‐system differentiation models outlines that steady state magma compositions at constantly replenished and erupting magmatic systems approximate simple fractional crystallization trends, due to short magma residence times. Open‐system magma evolution is associated with magma storage crystallinities that are lower than those associated with closed‐system scenarios. Accordingly, open‐system dynamics determine the efficient crystal‐melt separation toward the top of the reservoir, where eruptible melts continuously supply the ordinary activity. Conversely, a mush‐like environment constitutes the bottom of the reservoir, where poorly evolved magmas result from mixing events between mush residual melts and primitive magmas injected from deeper crustal levels. Plain Language Summary Volcanoes characterized by continuous eruptive activity are typified by constant replenishment of new magma, rising from deeper regions of the crust. The volcanic glass (supercooled silicate melt), represents the residual liquid of magma crystallization, and is found as the intracrystalline matrix of eruptive products. The study of its chemical composition may provide insight into the processes occurring at depths beneath the volcanic vent, where magma compositional changes result from crystallization and mixing with new magma rising from depth. We combine statistical analyses and analytical equations based on the chemical composition of the matrix glasses from Stromboli volcano, in order to constrain the processes which produce their chemical variations, identifying different environments where magmas are stored at depth. Our results also show that when magma is stored for a short period of time, the chemical changes to which the magma is subjected in a constantly replenished system are similar to those occurring in a system which is closed to new inputs of magma. Key Points The combination of multivariate statistics with geochemical modeling provides new constraints on magma differentiation processes Multivariate statistics based on trace elements allow better retrieval of petrological information than those based on major elements Magma differentiation in open systems approximates that occurring in closed systems when magma residence timescales are short