  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 Davies-Gaffney estimates. In this
paper, we introduce a new type of paraproduct operators that is constructed via
certain approximations of the identity associated to $L$. We show various
boundedness properties on $L^p(X)$ and the recently developed Hardy and BMO
spaces $H^p_L(X)$ and $BMO_L(X)$. In generalization of standard paraproducts
constructed via convolution operators, we show $L^2(X)$ off-diagonal estimates
as a substitute for Calder\'on-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 beyond Calder\'on-Zygmund theory, which
will be the subject of a forthcoming paper.
