Trailing 2-0 at the interval and 3-0 overall, Moura scored three times in the second half, including a last-gasp winner, to set up an all-English final against Liverpool in Madrid next month.
"It's impossible to explain what I'm feeling," Brazilian Moura told BT Sport after the game on Wednesday.
"We gave everything on the pitch and deserved this moment -- we are family.
"Football is amazing, its gives us moments like this. We cannot imagine. It's the best moment in my life, and in my career."
Such a scoreline seemed unfathomable when Ajax moved two goals ahead on the night inside the opening 35 minutes courtesy of Matthijs de Ligt's header and Hakim Ziyech's fine strike.
But Tottenham roared back after the interval with Moura scoring twice in four minutes to drag his side level on the night as Ajax began to wilt under constant waves of pressure.
And as the game moved into the final minute of stoppage time, Moura produced another moment of magic to send Tottenham through and leave this young Ajax side stunned.
But it was not just Dutch faces with tears streaming down their faces. Mauricio Pochettino, the Tottenham coach, cried with the sheer emotion of the achievement clearly too much for him to handle.
Wiping away the tears, Pochettino paid tribute to his players after their remarkable second-half showing.
"Thank you football," he told BT Sport. "My players are heroes. I'm so emotional now. It's an amazing moment, it's difficult to describe in words.
"When you work, when you feel the love, it's not a stress. It's a passion and we showed that passion."
Lightning strikes twice
For the second night in succession, football's ability to constantly shock and surprise came to the fore.
Even after the events at Anfield 24 hours earlier, where Liverpool had fought back from a three-goal first leg deficit to overcome Barcelona by winning 4-0 on the night, 4-3 on aggregate, few believed Tottenham could achieve similar feats especially after a nightmare opening period.
For Ajax, this will be a night that will live long in the memory, the night it somehow contrived to throw away the opportunity of a first Champions League final since 1996.
Quite how it managed to fritter away a two-goal lead on the night and a three-goal advantage overall will no doubt be discussed until the early morning across the bars of Amsterdam.
For 45 minutes at least, this was Ajax's opportunity. This team of young players had already surpassed all expectations in defeating Real Madrid and Juventus on its way to the last four of the competition.