Mauricio Pochettino's men headed to the Johan Cruyff ArenA 1-0 down after the first leg in north London and their task was made even more difficult after a disastrous first half.
Goals from Matthijs de Ligt and Hakim Ziyech saw the home side race into a 3-0 aggregate lead before the interval, with a final berth against Liverpool dangling agonisingly within their grasp.
However, Spurs were not to be denied, as a quickfire double from Moura before the hour mark turned the tie on its head, setting the stage for an incredible conclusion.
When shown the match-winning moment post-match, Moura attempted to sum up his feelings while clearly overcome with emotion.
Lucas Moura was shown the Brazilian commentary of his winner last night...— Goal (@goal) May 9, 2019
And this was his reaction 😢pic.twitter.com/AUo0pzez0n
The 26-year-old's outstanding performance capped a brilliant individual campaign which has seen him contribute 17 goals in over 40 appearances across all competitions.
Tottenham can now look forward to facing Liverpool at Wanda Metropolitano on June 1, but not before they wrap up their domestic campaign.
A positive result will also confirm their place in next season's Champions League, but they can also gain entry by winning the competition next month.
Spurs have not won a major trophy since lifting the League Cup in 2008 and their arrival in Madrid will mark the club's first appearance in a European Cup final.