Let X be a space of homogeneous type and let L be a sectorial operator with bounded holomorphic functional calculus on L^2(X) . We assume that the semigroup \{e^{-tL}\}_{t>0} satisfies the Davies–Gaffney estimates. In this paper, we introduce a new type of paraproduct operators that is constructed via certain approximations of the identity associated with L . We show various boundedness properties on L^p(X) and the recently developed Hardy and BMO spaces H^p_L(X) and {\rm BMO}_L(X) . Generalizing standard paraproducts constructed via convolution operators, we show L^2(X) off-diagonal estimates as a substitute for Calderón–Zygmund kernel estimates. As an application, we study differentiability properties of paraproducts in terms of fractional powers of the operator L . The results of this paper are fundamental for the proof of a T(1) -Theorem for operators that are beyond the reach of Calderón–Zygmund theory, which is the subject of a forthcoming paper.