We present a method for high-precision numerical evaluations of Lauricella functions whose indices are linearly dependent on some parameter ε in terms of their Laurent series expansions at zero. This method is based on finding analytic continuations of these functions in terms of Frobenius generalized power series. Being one-dimensional, these series are much more suited for high-precision numerical evaluations than multi-dimensional sums arising in approaches to analytic continuations based on re-expansions of hypergeometric series or Mellin–Barnes integral representations. To accelerate the calculation procedure further, the ε dependence of the result is reconstructed from the evaluations of given Lauricella functions at specific numerical values of ε, which, in addition, allows for efficient parallel implementation. The method has been implemented in the PrecisionLauricella package, written in Wolfram Mathematica language.