The third-tier League One side were the better team in the first half, exhausted in the second but then showed marvellous determination, and nous in the form of manager Keith Hill's canny use of substitutes like Steve Davies, whose equaliser in the third minute of stoppage time sealed the draw.
However, Moura was certainly Tottenham's brightest spark, proving a constant offensive threat as he grabbed his first goal for Spurs just before the hour mark.
Referee Bobby Madley pointed to the spot after Alli went down inside the penalty area under pressure from Harrison McGahey's challenge.
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Tottenham will not receive any home team privileges should they reach the FA Cup semi-finals at Wembley, it is understood.
It will be Tottenham's second successive replay after they had to settle for a draw in their fourth-round tie with League Two Newport County, who they defeated 2-0 at the second time of asking.
"I have not seen it again but I'm led to believe he is looking for it [but] we get too involved in controversy", he said.
"I've always dreamed of playing at Wembley and now I can say I've played there - if I'm fit and picked".
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Lucas Moura Team 1, Rochdale 1.
Eight-time FA Cup victor Tottenham, which left the likes of Harry Kane, Dele Alli and Christian Eriksen out of its starting lineup, stepped up a gear in the second half and Lucas Moura, signed from Paris Saint Germain in the midwinter transfer window, equalized in the 59th minute.
"It's a different competition", Pochettino said. Awaited.There you go. Still no Rochdale team by the way - maybe this is some good old fashioned Cup kidology! We have a lot of quality and a great structure and I think we can arrive at the end of the season with a trophy.
It seemed after such a good run of results against some of the biggest clubs in England and the European giants in Juventus, an away trip to Rochdale would be a stroll in the park.
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Of course, it's only Rochdale, but Spurs fans were encouraged and pretty impressed with what they saw from the Brazil worldwide and took to Twitter to share their thoughts.