Second-half substitute Quang The Nguyen scored with 15 minutes remaining of the second leg to stun the pre-tournament favourites and set up a final with three-times champions Thailand.
Run ends
Unbeaten in 20 matches, Singapore fought desperately for the win after their 0-0 first leg, with prolific striker Noh Alam Shah troubling the Vietnam defence throughout a tense encounter in the city-state.
Last year's top goalscorer and player of the tournament twice came close in the first half on Sunday and failed to connect on two dangerous crosses that flashed across the Vietnam goal.
The Lions paid for not capitalising on their pressure in the 75th minute when striker Le Cong Vinh rode two tackles before squaring the ball to Quang, who prodded home from five metres to trigger wild celebrations.
Vietnam will travel to Bangkok for the first leg of the final on December 24, with the return match in Hanoi on December 28.
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