Doan drove the ball into the bottom left corner, beating Vietnam's Russian-born goalkeeper Dang Van Lam who had guessed the right way after the spot kick was awarded in dramatic fashion shortly before the hour mark.
Appeals for a foul on Japan's Maya Yoshida in the box were initially dismissed by the referee, only for him to change his mind after a lengthy VAR review.
It was the second time VAR was used in the gripping clash, with an earlier Japan goal disallowed on review.
Vietnam coach Park Hang-seo said the VAR process could be quicker, though he did not dispute the penalty.
The technology, used at other tournaments, has been introduced to the Asian Cup for the first time and only from the last-eight stage.
"I hope the VAR problem is getting better," Park said, suggesting it should be introduced earlier so that referees were more comfortable with the technology.
Vietnam were quick on the counter attack but unable to get past a stubborn Japanese side who now face either China or Iran in the semi-finals.
"It was good that our players were able to keep a clean sheet," Japan coach Hajime Moriyasu told reporters. "As a team we earn another experience that will (help) our development."
Vietnam's Nguyen Cong Phuong missed an opportunity to break the deadlock in the 14th minute when he beat three defenders at Dubai's Al Maktoum Stadium but sent the ball wide.
Yoshida thought he put Japan ahead in the 24th minute when he headed in a cross, only for it to be disallowed for handball after a VAR review.
"I think my players did their best in this competition even though we lost today against Japan," Park said. "I'm really satisfied."
Japan midfielder Genki Haraguchi saw his effort from outside the box saved in the 68th minute, before the four-times champions survived a late onslaught.
Cong Phuong missed an opportunity to equalise in the 89th minute when he found himself unmarked just outside the box but he blasted the ball over.
(Reporting by Alexander Cornwell; Editing by Christian Radnedge)
