The direct growth of high-quality, large single-crystalline domains of graphene on a dielectric substrate is of vital importance for applications in electronics and optoelectronics. Traditionally, graphene domains grown on dielectrics are typically only ~1 μm with a growth rate of ~1 nm min-1 or less, the main reason is the lack of a catalyst. Here we show that silane, serving as a gaseous catalyst, is able to boost the graphene growth rate to ~1 μm min-1, thereby promoting graphene domains up to 20 μm in size to be synthesized via chemical vapour deposition (CVD) on hexagonal boron nitride (h-BN). Hall measurements show that the mobility of the sample reaches 20,000 cm2 V-1 s-1 at room temperature, which is among the best for CVD-grown graphene. Combining the advantages of both catalytic CVD and the ultra-flat dielectric substrate, gaseous catalyst-assisted CVD paves the way for synthesizing high-quality graphene for device applications while avoiding the transfer process.