Frenchman Frederic Piquionne and former Tottenham midfielder Kevin Prince Boateng, from the penalty spot, scored the extra-time goals to take Portsmouth back to the cup final for the second time in three years on the weekend they were consigned to the Coca-Cola Championship.
England goalkeeper David James also produced the save of the match to deny Peter Crouch at the end of normal time.
Grant said: "We had a belief and the courage to win this game, we knew we could do it.
"They (Spurs) are a good team with a great manager, but tactically we played very well. The players gave their hearts. It was not easy.
"Five players were not 100 per cent and some players did not play in their right position and we did not even have one defender on the bench. In spite of this we played football, scored two goals and I could not ask for more."
It was a good day, too, for Portsmouth administrator Andrew Andronikou, with the club guaranteed 900,000 in prize money for reaching the final, with another 1 million coming from television money and gate receipts.
For Grant, however, it was a fairytale after a season of turmoil in which the club was docked nine points for entering administration.
Grant said: "The way we did it after what has happened all this season and to see players give everything and not take the easy solution and not give up and to be in final, it is a crazy day. We could write a book about the many things against us this season."
Grant will now lead out his side against his former club Chelsea on 15 May.