The phenomenon of transient bursting, caused by additive noise in a set of two coupled FitzHugh-Nagumo oscillators, is studied by direct numerical integration and by measurements in the analog electronic circuit. In the parameter region where the unique global attractor of the deterministic system is the state of rest, introduction of low or moderate intensity fluctuations into the voltage dynamics results in the onset of a transient bursting state: sequences of intermittent bursts (patches of spikes), followed by ultimate relaxation to the equilibrium. Like genuine deterministic bursting, this behavior has its origin in the slow-fast character of the underlying dynamics. Trajectories that in the deterministic variant would converge to the state of rest can, under the action of noise, escape the local basin of attraction of the equilibrium and experience a bursting episode, before being dynamically reinjected into the region around the equilibrium. Under frozen parameter values and fixed noise intensity, the number of bursts preceding the ultimate decay strongly varies for different realizations of the additive random signal. The average duration of the transient bursting stage, bounded for weak noise, diverges when the intensity of fluctuations is raised. For sufficiently large ensembles of realizations, the lifetimes of transient bursting states, both in simulations and in the analog circuit, obey the exponential distribution. We relate this distribution to the probability for a stochastic trajectory to temporarily escape from the local basin of attraction of the equilibrium.