We demonstrate Schottky barrier engineering using PtOx/thin Pt Schottky contacts combined with edge termination using a high permittivity dielectric (ZrO2) field-plate for high-voltage vertical β-Ga2O3 diodes. A systematic study of baseline bare Pt/β-Ga2O3, PtOx/thin Pt/β-Ga2O3, and PtOx/β-Ga2O3 Schottky diode characteristics was performed, which revealed that the PtOx/thin Pt/β-Ga2O3 contact can combine the advantages of both PtOx and Pt, allowing better reverse blocking performance than plain metal Pt/β-Ga2O3 Schottky diodes and lower turn-on voltage than plain oxidized metal PtOx/β-Ga2O3 ones. Moreover, the thin Pt interlayer in the PtOx/thin Pt/β-Ga2O3 anode contact configuration, deposited by e-beam deposition, also provides plasma-free interface at the Schottky junction as opposed to the direct sputter deposited PtOx contacts of the PtOx/β-Ga2O3 diodes. We further implemented a high permittivity dielectric (ZrO2) field plate in PtOx/thin Pt/β-Ga2O3 diodes that assisted in edge-field management and enabled a breakdown voltage to ∼2.34 kV. These results indicate that the PtOx/thin Pt/β-Ga2O3 Schottky contact, combined with a high permittivity field-plate, will be promising to enable Schottky barrier engineering for high-performance and efficient vertical β-Ga2O3 power switches.