In the present paper, we evaluate two unified Eulerian integrals whose integrands involve the product of a general class of multivariable polynomials and the multivariable $ {\bf H}$-function. The results obtained here provide interesting unifications and extensions of several known [eg. [9], [11], [12], [18], [21], etc.] and new results. We have given two new special cases which are also sufficiently general in nature and of interest in themselves. As an application, one of the result has also been expressed as fractional integral which would provide useful generalization of known results in the theory of fractional calculus.