In this paper, we investigate the solution of multi-term time-space fractional partial delay differential-algebraic equations (MTS-FPDDAEs) with Dirichlet boundary conditions defined on a finite domain. We use Laplace transform method to give the solutions of multi-term time fractional delay differential-algebraic equations (MTS-FDDAEs). Then, the technique of spectral representation of the fractional Laplacian operator is used to convert the MTS-FPDDAEs into the MTS-FDDAEs. By applying our obtained solutions to the resulting MTS-FDDAEs, the desired analytical solutions of the MTS-FPDDAEs are obtained. Finally, we give the solutions of some special cases This work was supported by the Natural Science Foundation of China (NSFC) under grants 11871400. The work of J.J. Nieto has been partially supported by the Agencia Estatal de Investigacion (AEI) of Spain under Grant PID2020-113275GB-I00 and co-financed by European Community fund FEDER and by Xunta de Galicia, grant ED431C 2019/02. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature SI