Recent observations of coronal hole areas with the XRT and EIS instruments on board the Hinode satellite have shown with unprecedented detail the launching of fast, hot jets away from the solar surface. In some cases these events coincide with episodes of flux emergence from beneath the photosphere. In this Letter we show results of a three-dimensional numerical experiment of flux emergence from the solar interior into a coronal hole and compare them with simultaneous XRT and EIS observations of a jet-launching event that accompanied the appearance of a bipolar region in MDI magnetograms. The magnetic skeleton and topology that result in the experiment bear a strong resemblance to linear force-free extrapolations of the SOHO/MDI magnetograms. A thin current sheet is formed at the boundary of the emerging plasma. A jet is launched upward along the open reconnected field lines with values of temperature, density, and velocity in agreement with the XRT and EIS observations. Below the jet, a split-vault structure results with two chambers: a shrinking one containing the emerged field loops and a growing one with loops produced by the reconnection. The ongoing reconnection leads to a horizontal drift of the vault-and-jet structure. The timescales, velocities, and other plasma properties in the experiment are consistent with recent statistical studies of this type of event made with Hinode data.