We algorithmize the recent structural characterization for claw-free graphs by Chudnovsky and Seymour. Building on this result, we show that several domination problems are fixed-parameter tractable, and even possess polynomial-sized kernels, on claw-free graphs. To complement these results, we establish these problems are not fixed-parameter tractable on the slightly larger class of graphs that exclude $K_{1,4}$ as an induced subgraph. Our results provide a dichotomy for $K_{1,L}$-free graphs and show that the problems are fixed-parameter tractable if and only if $L\geq 3$. The algorithms we obtain generalize and unify several earlier algorithms for problems on claw-free graphs, and turn the existential decomposition by Chudnovsky and Seymour into an efficient constructive decomposition. (Joint work with Danny Hermelin, Erik Jan van Leeuwen and Gerhard Woeginger.)