We develop an algebraic approach to constructing short-time asymptotic expansions of solutions of a class of abstract semilinear evolution equations. The expansions are typically valid for both the solution of the equation and its gradient. We apply a perturbation approach based on the symbolic calculus of pseudo-differential operators and heat kernel methods. The construction is explicit and can be done to arbitrary order. All results are rigorously formulated in terms of Banach algebras. As an application we obtain a novel approach to finding approximate solutions of Markovian backward stochastic differential equations.