The heat produced during computation severely limits the performance of multi-/many-core processors. High-performance 3D-stacked processor-memory systems stack cores and main memory on a single die. However, 3D-stacked systems suffer more severe thermal issues than their non-stacked planar 2D counterparts. Consequently, the aggressive thermal throttling required for their thermally-safe operation limits the potential performance gains. Power budgeting is an effective thermal management technique that prevents thermal throttling in multi-/many-core processors by assigning a thermally-safe power budget to cores within the processors. State-of-the-art power budgeting techniques for 2D processors do not account for the vertical thermal coupling between the layers of the 3D-stacked system and will fail to prevent thermal throttling in them. Furthermore, estimating thermals for a 3D-stacked processor with power budgeting requires a finer-grained RC thermal model than non-stacked processors. This requirement inhibits the porting of existing power budgeting solutions for 2D processors to 3D-stacked processor-memory systems. This work is the first to present the linear algebra-based algorithmic time-invariant transformations required to enable power budgeting in 3D-stacked systems. Based on the transformations, we propose the first transient-temperature-aware power budgeting technique, 3D-TTP, for 3D-stacked systems. Detailed interval thermal simulations with the advanced CoMeT simulator designed for 3D-stacked systems also confirm no thermal violations with our 3D-TTP technique. 3D-TTP exhibits an average 11.41% speedup over the state-of-the-art reactive-based thermal management technique.