We derive the probability for a newly formed binary black hole (BBH) to undergo an eccentric gravitational wave (GW) merger during binary-single interactions inside a stellar cluster. By integrating over the hardening interactions such a BBH must undergo before ejection, we find that the observable rate of BBH mergers with eccentricity >0.1 at 10 Hz relative to the rate of circular mergers can be as high as ∼5% for a typical globular cluster (GC). This further suggests that BBH mergers forming through GW captures in binary-single interactions, eccentric or not, are likely to constitute ∼10% of the total BBH merger rate from GCs. Such GW capture mergers can only be probed with an N-body code that includes general relativistic corrections, which explains why recent Newtonian cluster studies have not been able to resolve this population. Finally, we show that the relative rate of eccentric BBH mergers depends on the compactness of their host cluster, suggesting that an observed eccentricity distribution can be used to probe the origin of BBH mergers.