All major biological processes start after protein molecules known as transcription factors detect specific regulatory sequences on DNA and initiate genetic expression by associating to them. But in eukaryotic cells, much of the DNA is covered by nucleosomes and other chromatin structures, preventing transcription factors from binding to their targets. At the same time, experimental studies show that there are several classes of proteins, called “pioneer transcription factors”, that are able to reach the targets on nucleosomal DNA; however, the underlying microscopic mechanisms remain not well understood. We propose a new theoretical approach that might explain how pioneer transcription factors can find their targets. It is argued that pioneer transcription factors might weaken the interactions between the DNA and nucleosome by substituting them with similar interactions between transcription factors and DNA. Using this idea, we develop a discrete-state stochastic model that allows for exact calculations of target search dynamics on nucleosomal DNA using first-passage probabilities approach. It is found that the target search on nuclesomal DNA for pioneer transcription factors might be significantly accelerated while the search is slower on naked DNA in comparison with normal transcription factors. Our theoretical predictions are supported by Monte Carlo computer simulations, and they also agree with available experimental observations.