We solve the semiclassical Einstein equations governing the dynamics of linear metric perturbations (of scalar, vector and tensor type) around de Sitter spacetime including the vacuum polarization effects of conformal matter fields. An order reduction method is used to eliminate the spurious solutions associated with higher order derivatives. Our exact (nonperturbative) solutions show that de Sitter spacetime is stable and a large time attractor. This extends some of the existing “no-hair” results for de Sitter spacetime.