This paper presents a modeling, analysis and simulation of a fluid-structure interaction model with an active thin structure, reproducing the behaviour of cilia or flagella immersed in a viscous flow. In the context of linear or nonlinear elasticity, the model is based upon the definition of a suitable internal Piola-Kirchoff tensor mimicking the action of the internal dyneins that induce the motility of the structure. In the subsequent fluid-structure interaction problem, two difficulties arise: on the one hand the internal activity of the structure which leads to more restricted well-posedness conditions and, on the other hand, the coupling conditions between the fluid and the structure that require a specific numerical treatment. In the context of numerical simulation, a weak formulation of the time-discretized problem is derived in functional frameworks that include the coupling conditions but, for numerical purposes, an equivalent formulation using Lagrange multipliers is introduced in order to get rid of the constraints in the functional spaces: this new formulation allows for the use of standard (fluid and structure) solvers, up to an iterative procedure. Numerical simulations are presented, including the beating of one or two cilia in 2d, discussing the competition between the magnitude of the internal activity and the viscosity of the surrounding fluid.