Responses of cells to signals are increasingly discovered to involve the binding of sequence specific transcription factors outside of known target genes. We wanted to determine to what extent the genome-wide binding and function of a transcription factor are shaped by the cell type versus the stimulus. To do so, we induced the Heat Shock Response pathway in two distant cell lines with two different stimuli and related the binding of its master regulator HSF1 to nascent RNA and chromatin accessibility. We show that HSF1 binding patterns robustly retain their identity under different magnitudes of activation so that common HSF1 binding is globally associated with stimulus-specific transcription outcomes. HSF1-induced increase in DNA accessibility was modest in scale but occurred predominantly at remote genomic sites. Apart from regulating transcription at existing elements including promoters and enhancers, responses to heat shock may directly engage inactive chromatin.