Substitute Mittag settled the contest shortly after her introduction (49), a strike bookended by Angerer's heroics to deny Trine Ronning and Solveig Gulbrandsen from 12 yards.
Germany's success was their eighth overall and the fourth time they have beaten Norway in the final.
And despite the starring role played by their goalkeeper, Silva Neid's side were decent value for their triumph in Solna at the end of a tournament in which they conceded just once.
That goal went to the Norwegians in their group stage victory, but they were on the back foot early on as Nadine Kessler's header came back off the bar via a touch from keeper Ingrid Hjelmseth.
Celia Da Mbabi was twice close, but Norway had the chance to snatch the lead when Cathrine Dekkerhus' tumble in the box earned a penalty.
Ronning opted for power and blasted the spot-kick down the middle, only for Angerer to stick out a leg and deflect the ball to safety.
Neid sent on Mittag at half-time, and the substitute's first significant contribution was to put the finishing touches to Da Mbabi's good work on the left.
Saskia Bartusiak's trip on Caroline Hansen then provided Norway with a second penalty, but Gulbrandsen fared no better than Ronning as Angerer's outstretched right arm intervened.
Ada Hegerberg later had a goal ruled out for offside, but Germany had several chances to make the game safe, Dzsenifer Marozsa firing just wide from 18 yards and Kessler side-footing against a post.