Stochastic chemical reaction networks constitute a model class to quantitatively describe dynamics and cell-to-cell variability in biological systems. The topology of these networks typically is only partially characterized due to experimental limitations. Current approaches for refining network topology are based on the explicit enumeration of alternative topologies and are therefore restricted to small problem instances with almost complete knowledge. We propose the reactionet lasso, a computational procedure that derives a stepwise sparse regression approach on the basis of the Chemical Master Equation, enabling large-scale structure learning for reaction networks by implicitly accounting for billions of topology variants. We have assessed the structure learning capabilities of the reactionet lasso on synthetic data for the complete TRAIL induced apoptosis signaling cascade comprising 70 reactions. We find that the reactionet lasso is able to efficiently recover the structure of these reaction systems, ab initio, with high sensitivity and specificity. With only < 1% false discoveries, the reactionet lasso is able to recover 45% of all true reactions ab initio among > 6000 possible reactions and over 102000 network topologies. In conjunction with information rich single cell technologies such as single cell RNA sequencing or mass cytometry, the reactionet lasso will enable large-scale structure learning, particularly in areas with partial network structure knowledge, such as cancer biology, and thereby enable the detection of pathological alterations of reaction networks. We provide software to allow for wide applicability of the reactionet lasso.