A multistage boiler heat exchange apparatus has a combustion furnace and at least one boiler set. The combustion furnace is used to produce a heat source and has a furnace base and a hot-air passage. The at least one boiler set is connected to the combustion furnace, and each has a preheater and a boiler. The preheater is deposited adjacent to the combustion furnace, and is connected to and communicates with the hot-air passage to adjust temperature of the heat source that enters the preheater. The boiler is an uprightly-deposited cylinder, is connected to the preheater, and has a conducting pipe and an exchange tube. The conducting pipe is deposited on and communicates with the boiler to enable the heat source to enter the boiler. The exchange tube is deposited in the boiler and has an exchange medium to exchange heat with the heat source in the boiler.