The constraint satisfaction problem (CSP) has important applications in computer science and AI. In particular, infinite-domain CSPs have been intensively used in subareas of AI such as spatio-temporal reasoning. Since constraint satisfaction is a computationally hard problem, much work has been devoted to identifying restricted problems that are efficiently solvable. One way of doing this is to restrict the interactions of variables and constraints, and a highly successful approach is to bound the treewidth of the underlying primal graph. Bodirsky & Dalmau [J. Comput. System. Sci. 79(1), 2013] and Huang et al. [Artif. Intell. 195, 2013] proved that CSP$(Γ)$ can be solved in $n^{f(w)}$ time (where $n$ is the size of the instance, $w$ is the treewidth of the primal graph and $f$ is a computable function) for certain classes of constraint languages $Γ$. We improve this bound to $f(w) \cdot n^{O(1)}$, where the function $f$ only depends on the language $Γ$, for CSPs whose basic relations have the patchwork property. Hence, such problems are fixed-parameter tractable and our algorithm is asymptotically faster than the previous ones. Additionally, our approach is not restricted to binary constraints, so it is applicable to a strictly larger class of problems than that of Huang et al. However, there exist natural problems that are covered by Bodirsky & Dalmau's algorithm but not by ours, and we begin investigating ways of generalising our results to larger families of languages. We also analyse our algorithm with respect to its running time and show that it is optimal (under the Exponential Time Hypothesis) for certain languages such as Allen's Interval Algebra.