T cells develop from multi-potent hematopoietic progenitors in the thymus and provide adaptive protection against pathogens and cancer. However, the emergence of human T cell-competent blood progenitors, and their subsequent specification to the T lineage, has been challenging to capture in real time. Here, we leveraged a pluripotent stem cell differentiation system to understand the transcriptional dynamics and cell fate restriction events that underlie this critical developmental process. Time-resolved single cell RNA sequencing revealed that cell-cycle exit, downregulation of the multipotent hematopoietic program, and upregulation of >90 lineage-associated transcription factors all occur within a highly co-ordinated and narrow developmental window. Computational gene-regulatory network inference elucidated the transcriptional logic of T lineage specification, uncovering an important role for YBX1. We mapped the differentiation cell fate hierarchy using transcribed lineage barcoding and mathematical trajectory inference and discovered that mast and myeloid potential bifurcate from each other early in haematopoiesis, upstream of T lineage restriction. Collectively, our analyses provide a quantitative, time-resolved model of human T cell specification with relevance for regenerative medicine and developmental immunology.