For much of the contest, Pakistan looked favourites to progress from Thursday's last-four encounter, but three successive sixes from Wade in the penultimate over swung the match in Australia's favour.
Pakistan would have had no complaints with their innings, 71 runs initially being put on for the first wicket before Babar Azam was out for 39.
To their credit, Warner (49) and Mitchell Marsh (22) got Australia back into the contest, the Baggy Greens reaching 89-4 at the halfway stage.
However, Shadab Khan was sensational with the ball, taking three wickets with his first 13 balls, and Pakistan appeared to be well on their way to the final when Shabah removed Glenn Maxwell for his fourth wicket to leave Australia needing 81 from 46.
That was until Wade came to the crease, the wicket-keeper biding his time alongside Marcus Stoinis (40) before securing victory in a remarkable penultimate over.
After being put down on the boundary, Wade hit the next three deliveries for six, ending the contest with a majestic 41 from just 17 balls.
Australia, having won with eight deliveries to spare, will face New Zealand in the final after the Black Caps defeated England on Wednesday. body check tags ::