Ian Henderson gave the hosts a surprising lead on the stroke of half-time by finishing off a counter-attack.
New Spurs signing Lucas Moura scored in his first start for the club, equalising on the 59th minute mark after he latched onto Moussa Sissoko's pass.
"The moment the ball (Steven Davies' goal) hit the back of the net was a reward," said Rochdale manager Keith Hill.
"The performance deserved the opportunity to at least go to Wembley in a replay," said Hill in his post-match press conference.
"I thought the first half was magnificent with the way we took the game to the opposition. Why not take them on, why not try and win it? And I thought we gained their respect.
"It was tremendous for us to score the first goal and go into half-time winning – it was a marvellous feeling for the players. In the second half, I thought the players were magnificent in their response to both goals.
"For us to get that late equaliser, wow! That was raw emotion, not just from me, but from the players, the staff and from the supporters as well. I'm really pleased that the players have got their rewards and are going to Wembley."