Anyons are quasiparticles that keep a robust memory of particle exchanges via a braiding phase factor. When an anyon excitation is emitted toward a quantum point contact (QPC) in a fractional quantum Hall (FQH) fluid, this memory translates into tunneling events that may occur long after the anyon excitation has exited the QPC. In this work, we used triggered anyon pulses incident on a QPC in a filling factor ν = 1/3 FQH fluid to investigate anyon tunneling in the time domain. We observed that braiding increases the tunneling timescale, which is set by the temperature and the anyon scaling dimension that characterizes the edge-state dynamics. Our experiment introduces time-domain measurements for characterizing the braiding phase and scaling dimension of anyons.