We propose a systematic generating procedure to construct free Lagrangians for massive, massless and partially massless, totally-symmetric tensor fields on $AdS_{d+1}$ starting from the BRST Lagrangian description of massless fields in the flat ambient space $\mathbb{R}^{d,2}$. A novelty is that the Lagrangian is described by a $d+1$ form on $\mathbb{R}^{d,2}$ whose pullback to $AdS_{d+1}$ gives the genuine Lagrangian defined on anti de Sitter spacetime. Our derivation uses the triplet formulation originating from the first-quantized BRST approach, where the action principle is determined by the BRST operator and the inner product of a first-quantised system. In this way we build, in a manifestly $so(2,d)$-covariant manner, a unifying action principle for the three types of fields mentioned above. In particular, our derivation justifies the form of some actions proposed earlier for massive and massless fields on AdS. We also give a general setup for ambient Lagrangians in terms of the respective jet-bundles and variational bi-complexes. In particular we introduce a suitable ambient-space Euler-Lagrange differential which allows to derive the equation of motion ambiently, i.e. without the need to explicitly derive the respective spacetime Lagrangian.