PSG routed Inter Milan 5-0 in Munich as Desire Doue's double, as well as goals from Achraf Hakimi, Khvicha Kvaratskhelia and Senny Mayulu, secured a historic treble for the French league and cup winners.
After the game, PSG supporters unveiled a tifo paying tribute to Enrique's daughter Xana, who died in 2019 at the age of nine from cancer.
Enrique said: "I'm very happy. It was very emotional at the end with the banner from the fans for my family.
"But I always think about my daughter."
The Spaniard, who played 62 times for his senior national team between 1991 and 2002, joined PSG last July and has overseen the development of a brilliant young team following the departure of star striker Kylian Mbappe to Real Madrid.
It was the 55-year-old's second treble as a manager after winning the Champions League, La Liga and Copa del Rey at Barcelona in 2015.
"Since day one, I said I wanted to win important trophies, and Paris had never won the Champions League," Enrique added.
"It's a great feeling to make many people happy ... Inter are a great side, but we were fantastic in our pressing.
"Every player improved this season, and so did the team.
"We prepared the final with the necessary calm to put the excitement in the right place. Now we can enjoy it; we can take the trophy back to Paris.
The Eiffel Tower was lit up in the red and blue of PSG to mark their triumph, only the second by a French club after Marseille in 1993.
Enrique has become the seventh coach to win the competition with two different teams and follows in the footsteps of greats like Carlo Ancelotti, Pep Guardiola and Jose Mourinho.
He is the 21st coach to win multiple European Cups.
