The pair were third after the opening three rounds but produced two outstanding dives to finish off the competition, scoring more than 75 points for both efforts.
The Great Britain duo scored an impressive 321.06 points to finish well ahead of silver medallists Tania Cagnotto and Maicol Verzatto of Italy, who scored 308.70 points.
Russia’s Nadezhda Bazhina and Nikita Shleikher were third with 305.28.
Daley said afterwards: “I don’t really know what just happened. Grace and I were put together 10 days ago and we’ve done about five sessions together so I’m over the moon to come away as European champions on a board that I never train.
“It was great to have the home crowd, our families in the audience. It’s been really fun although I was terrified before starting the competition.
Daley will compete in the men’s 10m synchro event on Thursday with Plymouth Diving’s Dan Goodfellow and then take part in the individual platform competition on Sunday.