We study the problem of efficiently generating high-quality and diverse content in games. Previous work on automated deckbuilding in Hearthstone shows that the quality diversity algorithm \mbox{MAP-Elites} can generate a collection of high-performing decks with diverse strategic gameplay. However, MAP-Elites requires a large number of expensive evaluations to discover a diverse collection of decks. We propose assisting MAP-Elites with a deep surrogate model trained online to predict game outcomes with respect to candidate decks. MAP-Elites discovers a diverse dataset to improve the surrogate model accuracy, while the surrogate model helps guide MAP-Elites towards promising new content. In a Hearthstone deckbuilding case study, we show that our approach improves the sample efficiency of MAP-Elites and outperforms a model trained offline with random decks, as well as a linear surrogate model baseline, setting a new state-of-the-art for quality diversity approaches in automated Hearthstone deckbuilding. We include the source code for all the experiments as supplemental material.