Two-dimensional hexagonal boron nitride (h-BN) has excellent and useful mechanical and thermal properties, and can be used as a novel filler to enhance the thermal conductivity of polymer composites. We prepared exfoliated h-BN nanoplatelets (BNNP) with hydroxyl functional groups using a hydroxide-assisted ball milling process, and demonstrated the enhanced thermal conductivity of its epoxy nanocomposites. The prepared BNNP are highly soluble and retain their in-plane structure. The dielectric constant and dielectric loss of the nanocomposites increase with the addition of BNNP owing to interfacial polarization by the large surface area of BNNP and the ionic and electronic relaxation polarizations by the surface functional groups on the BNNP. Notably, the thermal conductivity of the epoxy nanocomposites with 10% mass fraction of BNNP was 0.57 W/m·K, which is 2.85-times higher than that of neat epoxy. Finally, compared with neat epoxy, BNNP nanocomposite exhibits excellent heat dissipation capability in flat chip packaging. This enhanced performance is ascribed to the high quality and dispersion of the BNNP and their strong interfacial bonding with the epoxy matrix, which is produced by the hydroxyl functional groups on the BNNP. The overall results suggest that BNNP nanocomposites have strong potential for application as electronic packaging materials.