Jordan had been ahead when Baha Abdel-Rahman's sweetly struck free-kick gave Vital Borkelmans' side the initiative just before half-time, but the second period saw a resurgent Vietnam take control of the game.
Nguyen Cong Phuong's equaliser rattled Jordan, who then withstood successive waves of Vietnam pressure to take the game into a goalless period of extra-time.
Baha' Seif rattled the crossbar with Jordan's second penalty before Van Lam saved from Ahmed Salah, and Bui Tien Dung I then rolled in the winning strike to send Vietnam into the last eight.
Vietnam coach Park Hang-seo said he expected a physical encounter and his side shirked none of the early duels while containing the threat posed by Yousef Rawshdeh in the Jordan attack
The Golden Dragons carved out their best chance of the first half when Doan Van Hau played a fast one-two with Nguyen Quang Hai before lashing a low shot towards goal, but experienced Jordan goalkeeper Amer Shafi got down well to parry the ball away from danger.
After Do Hung Dung was penalised for his high challenge on Salem Al Ajalin, Abdel-Rahman stepped up and curled the ensuing free-kick from the side of the penalty area into the far corner of the net to put the Brave Gentlemen 1-0 up at the interval.
Just six minutes into the second half Vietnam drew level when Nguyen Trong Hoang's inswinging cross beat a line of Jordan defenders and Cong Phuong slid in to meet it with a volley that flew over Shafi's head before he could react.
The equaliser galvanised Vietnam who might have taken the lead had Phan Van Duc connected cleanly with his shot when he raced through on goal 10 minutes later, and Dung drilled a low swerving effort that Shafi could only punch clear as extra-time loomed.
Yasen Bakheet's looping volley over the crossbar was the best either side could muster in extra time, and when it mattered most it was Van Lam who held his nerve, diving to his right to save Salah's penalty and clinch victory for Vietnam in the shoot-out.