We use QCD Laplace sum-rules to predict masses of open-flavour heavy-light hybrids where one of the hybrid's constituent quarks is a charm or bottom and the other is an up, down, or strange. We compute leading-order, diagonal correlation functions of several hybrid interpolating currents, taking into account QCD condensates up to dimensionsix, and extract hybrid mass predictions for all J P ∈ {0 ± , 1 ± }, as well as explore possible mixing effects with conventional quark-antiquark mesons. Within theoretical uncertainties, our results are consistent with a degeneracy between the heavy-nonstrange and heavystrange hybrids in all J P channels. We find a similar mass hierarchy of 1 + , 1 − , and 0 + states (a 1 + state lighter than essentially degenerate 1 − and 0 + states) in both the charm and bottom sectors, and discuss an interpretation for the 0 − states. If conventional meson mixing is present the effect is an increase in the hybrid mass prediction, and we estimate an upper bound on this effect.