We present observations of ASASSN-19dj, a nearby tidal disruption event (TDE) discovered in the post-starburst galaxy KUG 0810+227 by the All-Sky Automated Survey for Supernovae (ASAS-SN) at a distance of d similar or equal to 98 Mpc. We observed ASASSN-19dj from -21 to 392 d relative to peak ultraviolet (UV)/optical emission using high-cadence, multiwavelength spectroscopy and photometry. From the ASAS-SN g-band data, we determine that the TDE began to brighten on 2019 February 6.8 and for the first 16 d the rise was consistent with a flux proportional to t(2) power law. ASASSN-19dj peaked in the UV/optical on 2019 March 6.5 (MJD = 58548.5) at a bolometric luminosity of L = (6.2 +/- 0.2) x 10(44) erg s(-1). Initially remaining roughly constant in X-rays and slowly fading in the UV/optical, the X-ray flux increased by over an order of magnitude similar to 225 d after peak, resulting from the expansion of the X-ray emitting region. The late-time X-ray emission is well fitted by a blackbody with an effective radius of similar to 1 x 10(12) cm and a temperature of similar to x 10(5) K. The X-ray hardness ratio becomes softer after brightening and then returns to a harder state as the X-rays fade. Analysis of Catalina Real-Time Transient Survey images reveals a nuclear outburst roughly 14.5 yr earlier with a smooth decline and a luminosity of L-V >= 1.4 x 10(43) erg s(-1), although the nature of the flare is unknown. ASASSN-19dj occurred in the most extreme post-starburst galaxy yet to host a TDE, with Lick H delta A = 7.67 +/- 0.17 angstrom.