Invagination of organ placodes converts flat epithelia into three-dimensional organs. Cell tracing in the Drosophila tracheal placode revealed that, in the 30-minute period before invagination, cells enter mitotic quiescence and form short rows that encircle the future invagination site. The cells in the rows align to form a smooth boundary (;boundary smoothing'), accompanied by a transient increase in myosin at the boundary and cell intercalation oriented in parallel with the cellular rows. Cells then undergo apical constriction and invaginate, followed by radially oriented mitosis in the placode. Prior to invagination, ERK MAP kinase is activated in an outward circular wave, with the wave front often correlating with the smoothing cell boundaries. EGFR signaling is required for myosin accumulation and cell boundary smoothing, suggesting its propagation polarizes the planar cell rearrangement in the tracheal placode, and coordinates the timing and position of intrinsic cell internalization activities.