To tackle the challenge of modeling high-dimensional right-censored trial data, we introduce latently unobserved failure times that define the full likelihood of the data set. Unfortunately, integrals over these latent time variables give rise to intractable forms. To mitigate this difficulty, Monte Carlo estimates can be used. However, this introduces undesirable noise into the results, complicating gradient-based optimization and reducing scalability. Instead, we establish a transformation that scales the latent failure times to the unit exponential scale, which leads to a form of the censored likelihood that can be parsed as a continuous path integral. We then use a steepest descent argument to determine the saddle point of this integral deterministically and compute a Gaussian approximation to the full integral using a second-order Taylor expansion about its saddle point. The surrogate likelihood is fully differentiable and depends only on the output from the marginal hazard networks and a copula parameterization of the data. Through the application of automatic differentiation, it is possible to determine the gradients corresponding to all model parameters directly, without the need for the stochastic approximations. By using a suitable parameterization, we can learn a regular vine copula, where the pair-copulas are represented by neural networks. By reclustering the data in recursive saddle points, we can decrease the complexity of copula learning dramatically, and often we can compute closed-form conditional saddle points. The neural marginal hazard networks and vine correlation weights can then be optimized end to end with a preconditioned gradient optimizer. In inference, the (deterministic) saddle point is treated as the most likely latent configuration of the model, and it allows for survival and conditional survival predictions. The construction and use of the differentiable, fixed surrogate for the integrals propagates its influence to all node distributions and allows for scale to potentially hundreds of dimensions with complicated dependencies in censored survival data. The framework of deep generative copulas for probabilistic survival analysis thus combines the advantages of sampling approaches with fully deterministic learning and provides the most principled way to perform inference.