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The present work is devoted to proposing a low-cost spectral method based on the modified hat functions for solving fractional delay differential equations. The fractional derivative is considered in the sense of Caputo. In order to solve the considered problem, the existing functions in it are approximated using the basis functions. By employing some important properties of the basis functions, Caputo derivative and Riemann–Liouville fractional integral, the main problem is transformed into some systems of nonlinear algebraic equations including two unknown parameters. This procedure mainly simplifies the problem and gives its approximate solution after solving the resulting systems. In addition, the computational complexity of the derived system is investigated. An error analysis is discussed to show the convergence order of the method. Finally, the suggested technique is applied to some sample problems with the aim of checking its validity and accuracy. • A numerical approach for solving fractional delay differential equations is proposed. • The method uses modified hat functions, being efficient and high performant. • We reduce the solution of the problem to a system of nonlinear algebraic equations. • The complexity of the obtained system is analyzed. • An error estimate of the method is proved. [ABSTRACT FROM AUTHOR]