A class of finite-dimensional algebras with operator-valued structural constants is introduced and studied. Particular attention is given to a two-dimensional commutative associative operator algebra whose multiplication is determined by two commuting linear operators. Basic algebraic properties of this algebra are established, and a theory of monogenic functions with operator-valued components is developed. Operator analogues of the Cauchy–Riemann conditions are obtained. It is shown that the components of monogenic functions satisfy a second-order linear operator-differential equation. This approach provides a constructive method for obtaining solutions of various classes of partial differential, integro-differential, and functional equations.