Sweden are the second-most successful nation in curling at the Winter Olympics, but they had never previously won a mixed doubles gold medal, with Almida de Val and Oskar Eriksson's bronze their best previous effort in the format.
After limiting the United States pair of Korey Dropkin and Cory Thiesse to one point in the first end, Wranaa and brother Rasmus took a 2-1 lead when Isabella deftly nestled a stone against another on the button.
The pairs traded a point apiece in the next two ends to leave Sweden 3-2 up at the halfway mark.
In identical fashion to their 9-3 win over Britain in the semi-final, Sweden deployed their powerplay with the score level at 3-3 in the sixth end.
However, unlike Monday's match, in which they grabbed a game-winning five points, the tactic yielded just the one point, and the US hit back with two in the penultimate end to take a 5-4 lead.
The momentum seemed to be shifting in favour of the US, but Dropkin and Thiesse both slightly overcooked shots before the ever-reliable Isabella Wranaa sealed the win on the final stone.
The Wranaas' victory moves Sweden level with Canada on 12 total medals, though Canada have one more gold.
Earlier on Tuesday, Italy's Amos Mosaner and Stefania Constantini saw out a tense 5-3 win over Britain's Jennifer Dodds and Bruce Mouat to add the bronze medal to their gold from four years ago.