Spatial architectures are more efficient than traditional Out-of-Order (OOO) processors for computationally intensive programs. However, spatial architectures require mapping a program, either statically or dynamically, onto the spatial fabric. Static methods can generate efficient mappings, but they cannot adapt to changing workloads and are not compatible across hardware generations. Current dynamic methods are adaptive and compatible, but do not optimize as well due to their limited use of speculation and small mapping scopes. To overcome the limitations of existing dynamic mapping methods for spatial architectures, while minimizing the inefficiencies inherent in OOO superscalar processors, this paper presents DynaSpAM (Dynamic Spatial Architecture Mapping), a framework that tightly couples a spatial fabric with an OOO pipeline. DynaSpAM coaxes the OOO processor into producing an optimized mapping with a simple modification to the processor's scheduler. The insight behind DynaSpAM is that today's powerful OOO processors do for themselves most of the work necessary to produce a highly optimized mapping for a spatial architecture, including aggressively speculating control and memory dependences, and scheduling instructions using a large window. Evaluation of DynaSpAM shows a geomean speedup of 1.42x for 11 benchmarks from the Rodinia benchmark suite with a geomean 23.9% reduction in energy consumption compared to an 8-issue OOO pipeline.