Diego Godin looked to have won the cup for Diego Simeone's side with his first-half goal but Sergio Ramos levelled matters with just over 90 seconds of normal time remaining.
Real then ran riot in extra-time, with Gareth Bale, Marcelo and Cristiano Ronaldo all on target as Los Blancos finally realised their Decima dream.
"It's the biggest trophy in club football; an incredible feeling," the German told the DFB's official website. "One that makes me proud.
"Inside I felt a great joy. We were dead but then Sergio Ramos saved us once again."
Khedira was a surprise starter in Lisbon, having been selected ahead of Asier Illarramendi to replace the suspended Xavi Alonso in midfield - in spite of the fact that the Germany international has only just returned to action after six months out with a knee injury.
The 27-year-old only last 59 minutes before being replaced by Isco but he insists that he will be match fit by the time the World Cup gets under way.
"I think it was self-evident that [a lack of game time] did not allow me to be at my best on the pitch," he admitted. "That's why I had to go off.
"But I think if I had not been ready then I would not have played from the start. It was a great sign of confidence in me by the coach [Carlo Ancelotti] that I was allowed to play without having built up any rhythm. That's the only thing missing now.
"But, before the World Cup, [Germany] have two or three games and that will be enough for me. I can reassure everyone of that."