Speaking to the club's official channels afterwards, the Spaniard did not hide his emotion at pulling on the white shirt for the first time.
"It's a matter of pride. I don't think many people can say they've worn this shirt. We've seen great legends, it's a club with a lot of history, the biggest in Europe," he said.
"Being here today is a privilege, an honour, and above all a reward for all the effort I've had to put into my career. It's been a long wait, it felt long, but I'm here now."
Cucurella, who joined for a fee in the region of €55-60 million after completing his move from Chelsea, described watching Real Madrid from the outside for years before getting his own chance to wear the badge.
"As a player, as a kid who starts out playing football, having the chance to play for a club like this is very hard to turn down. The opportunity came along and I didn't have a single doubt," he said.
"As a spectator, I've seen all those magical nights, those comebacks at the Bernabéu, all the Champions Leagues they've won. Having the chance to experience it first-hand and be part of it is a huge responsibility. But also a really exciting and important challenge."
Asked to describe himself as a player and person, the 27-year-old left-back pointed to his work rate and the dressing room he is joining.
"Intensity, character, joy, a bit of everything. I'm a very hard-working person," he said. "With players of this talent, as long as we stay united and are a great team both on and off the pitch, I think we can achieve great things."
On his first impressions of working under Mourinho, Cucurella struck an upbeat tone.
"A lot of trust and a natural approach. We have to do what we know how to do on the pitch, and above all enjoy it," he said. "I think what we all want is to win and for it to be a great year."
The former Chelsea full-back, who signed a six-year contract running to June 2032, also spoke about settling in among his new teammates on his first day at Valdebebas.
"Everything's going really well. I was already looking forward to meeting my teammates. I've spoken to some who'd messaged me. The welcome has been great," he said. "It's been a hard, long day, but I'm settling in and training little by little, because the good stuff is coming soon."
Cucurella arrives at the Bernabéu with 163 appearances for Chelsea behind him, having won the 2025 Club World Cup and a Europa Conference League during his time in west London, and becomes Mourinho's first confirmed signing of the summer as Real Madrid continue reshaping their squad.