Despite a major blunder from Paul Robinson in the first half, Tottenham looked to have put their name in the hat for the fourth round after Dimitar Berbatov netted a brace for the home side.
But Hunt shocked the home crowd when he fired home from close range to put a huge dent in Juande Ramos' attempts to win some silverware this season.
Reading took the lead in bizarre circumstances on 25 minutes following Robinson's blunder.
Hunt's free-kick appeared to be dealt with by the England keeper but he bizarrely took the ball over his own line - much to the amazement of the crowd at White Hart Lane.
But the home side hit back and grabbed an equaliser on 28 minutes.
Aaron Lennon's pass picked out Berbatov, who found himself totally unmarked in the box and showed all his quality once again when he fired an unstoppable shot past Adam Federici into the top left-hand corner.
Tottenham could have added a second on 34 minutes when Robbie Keane's back-heel fell to Jermaine Jenas, who raced into the box but saw his shot beaten out by the legs of Federici.
Jenas had another golden chance on the stroke of half-time when Lennon played him through on goal, but he was once again denied by Federici.
The home side were rewarded when Berbatov added his second of the game in the 50th minute.
Michael Dawson's long ball picked out Keane, who was brought down inside the box by Liam Rosenior.
Berbatov made no mistake by striking the spot-kick past Federici.
But despite making eight changes to their team, Reading refused to lie down and grabbed an equaliser on 78 minutes.
Leroy Lita's shot was kept out by Robinson and Hunt reacted quickest to knock the rebound home.
Tottenham's misery was compounded when Tom Huddlestone was sent off on 81 minutes following an apparent headbutt on Bobby Convey.