Wycombe had led 2-0 and 3-1 at White Hart Lane before Spurs' late comeback sent the League Two side crashing out of the tournament in heartbreaking fashion.
Despite Mauricio Pochettino making a number of changes, Spurs were still favourites to prevail with ease, but they were given an early warning when Paul Hayes headed against the crossbar in the first minute.
Son could have given Spurs the lead when he capitalised on a mistake by Jamal Blackman but he managed to miss the target, and the home side were soon made to pay.
Falling behind resulted in a positive response from Spurs but shortly before the half-hour mark, Son could not find a way past Blackman, before the South Korean attacker fired over the crossbar.
Nine minutes before the break, Wycombe were in dreamland when Sam Wood was tripped inside the penalty area and Hayes was able to convert into the net from 12 yards.
Pochettino introduced Vincent Janssen at the break, but Son remained Spurs' best chance of a goal and on the hour mark, he obliged with a fine finish from an acute angle.
Four minutes later, Spurs were back on level terms through Janssen, who slotted into the net from the penalty spot after he had been brought down in the area.
Spurs had the momentum heading into the closing stages but they were dealt a blow with 16 minutes left when having made each of their three substitutions, Kieran Tripper was forced off injured.
However, Spurs were not finished and with three minutes left, they equalised for the second time in the match when a mistake in the Wycombe defence allowed Dele Alli the chance to score from a narrow angle.
It appeared that the teams would require a replay but in the final seconds of the game, Son played a one-two with Janssen before sending the ball past Blackman to earn Spurs a place in the last 16.