Dortmund, unfancied against the Spanish giants, dominated the first half and went close on several occasions, but failed to land a killer blow.
Dortmund striker Niclas Fuellkrug hit the post and Karim Adeyemi had two clear chances to open the scoring, but Real held on.
The Spanish giants then scored twice in 10 minutes late in the second half to lift the European Cup for the 15th time in their history.
Dani Carvajal headed in from a corner before former Dortmund midfielder Jude Bellingham teed up Vinicius Junior, who had troubled Dortmund all night, to fire into the far corner.
Speaking with Germany's ZDF network, Hummels said Dortmund "played a great game" and he was "really proud of the team."
"We showed a lot of courage, heart and footballing skill. We missed out on scoring and then conceded the goal. That's how they always do it," the 35-year-old said.
"That shows their quality, but it also took a bit of luck today."
Dortmund coach Edin Terzic said his side "deserved a little bit more" but lacked "killer instinct".
"Above all in the first half we had the feeling that we had them. From the first second we showed them that we're here to win it."
Dortmund goalkeeper Gregor Kobel told ZDF: "At the moment we're just disappointed - the disappointment is huge."
"You don't get too many chances against Real Madrid, we had our chances and should have made something of them once or twice.
"Nevertheless, reaching the final is a huge success."
