Particles having finite mass and size advected in open chaotic flows can form attractors behind structures. Depending on the system parameters, the attractors can be chaotic or nonchaotic. But, how robust are these attractors? In particular, will small, random perturbations destroy the attractors? Here, we address this question by utilizing a prototype flow system: a cylinder in a two-dimensional incompressible flow, behind which the von Kármán vortex street forms. We find that attractors formed by inertial particles behind the cylinder are fragile in that they can be destroyed by small, additive noise. However, the resulting chaotic transient can be superpersistent in the sense that its lifetime obeys an exponential-like scaling law with the noise amplitude, where the exponent in the exponential dependence can be large for small noise. This happens regardless of the nature of the original attractor, chaotic or nonchaotic. We present numerical evidence and a theory to explain this phenomenon. Our finding makes direct experimental observation of superpersistent chaotic transients feasible and it also has implications for problems of current concern such as the transport and trapping of chemically or biologically active particles in large-scale flows.