We construct a dg-enhancement of Webster’s tensor product algebras that categorifies the tensor product of a universal $\mathfrak{sl}_2$ Verma module and several integrable irreducible modules. We show that the blob algebra acts via endofunctors on derived categories of such dg-enhanced algebras in the case when the integrable modules are two-dimensional. This action intertwines with the categorical action of $\mathfrak{sl}_2$. From the above we derive a categorification of the blob algebra.