Di Maria's 53rd minute header handed Juventus a slim advantage going into the return leg in Germany next week.
The 35-year-old scored a hattrick in the previous round against Nantes and now has eight goals and four assists in 24 appearances for Juventus this season.
Di Maria signed a one-year contract when he moved to the Allianz Stadium from Paris Saint Germain on a free transfer last summer.
After a slow start not helped by niggling injuries, the World Cup winner is now back to his very best.
"I'm happy to have helped the team, we are doing a good job and we must continue like this," he told Sky. "The important thing was to win, and we did it.
"We will do our best [to reach the final of this competition]. We will try to win both the Coppa Italia and the Europa League.
"We're talking, I'm happy here, and I'm finally feeling the affection of the people after a difficult start."
Juventus manager Massimiliano Allegri was equally effusive about Di Maria's performance.
"Tonight, [Di Maria] scored a wonderful goal but, above all, he gave an extraordinary performance.
"We played a very good match tonight, with great intensity and good technically. The only thing we missed too many times was the last pass. We could have chosen better and that made the difference.
"In Freiburg, it won't be easy, but the thing that makes me think positive is that they will give us spaces, and we will have to exploit them."
