Context. Asteroseismic investigations of solar-like oscillations in giant stars allow for the derivation of their masses and radii. For members of open clusters, this allows us to obtain an age estimate of the cluster that is expected to be identical to the age estimate given by the colour-magnitude diagram, but independent of the uncertainties that are present for that type of analysis. Thus, a more precise and accurate age estimate can be obtained. Aims. We aim to identify and measure the asteroseismic properties of oscillating giant members of the open cluster NGC 6866 and utilise them for a cluster age estimate. Model comparisons also allow constraints to be placed on the stellar physics. Here, we investigate the efficiency of convective-core overshoot during the main sequence evolution, which has a significant influence on the age estimations for these relatively massive giants. The effects of rotation and core overshoot are similar, but not identical, and so, we also investigated the potential of our measurements to distinguish between these effects. Methods. We identified six giant members of NGC 6866 via photometry, proper motions, and parallaxes from Gaia, and spectroscopic literature measurements. These were combined with asteroseismic measurements, which we derived using photometric data from the Kepler mission for five of the stars. Comparisons to stellar-model isochrones constrained the convective-core overshoot and enable a more precise and accurate age estimate than previously possible. Results. A significant amount of differential reddening is found for NGC 6866. Asteroseismology establishes the helium-core burning evolutionary phase for the giants, which have a mean mass of 2.8 M_⊙. Their radii are significantly smaller than predicted by current 1D stellar models unless the amount of convective-core overshoot on the main sequence is reduced to α_ov ≤ 0.1 ⋅ H_p in the step-overshoot description. Our measurements also suggest that rotation has affected the evolution of the stars in NGC 6866 in a way that is consistent with 3D simulations, but not with current 1D stellar models. The age of NGC 6866 is estimated to be 0.43 ± 0.05 Gyr, which is significantly younger and more precise than most previous estimates. Conclusions. We derive a precise cluster age while constraining convective-core overshooting and the effects of rotation in the stellar models. A comparison to age estimates from machine learning methods of the same and similar giant stars uncovers potential biases for automated asteroseismic and non-asteroseismic age estimates of helium-core burning stars.