Italy-based winger Adnan placed his strike high and to the left of Vietnam goalkeeper Dang Van Lam to give 2007 winners Iraq a barely-deserved victory in their opening Group D encounter in Abu Dhabi.
"It is very important to get three points in the group stage of this tournament," said Iraq coach Srecko Katanec.
"It was tough at times, but I am sure we will develop as the competition goes on. If I am honest, we also need to be even better than we were in the second half which was obviously better than the first.
"I saw today who can and cannot play going forward, who is in a good condition and who is not."
In a game strewn with defensive errors, Vietnam took the lead in the 24th minute when Ali Faez back-heeled the ball into his own goal after a mix-up with goalkeeper Jalal Hassan.
Mohanad Ali levelled for Iraq 11 minutes later when he dispossessed Que Ngoc Hai 25 metres from goal before driving into the penalty area and slotting past Van Lam.
Vietnam reclaimed the lead three minutes before the break when Nguyen Trong Hoang's low shot was parried by Hassan and Nguyen Cong Phuong tapped in the rebound.
Iraq, however, were level again on the hour mark when Humam Tariq slid home from close range, making the most of Vietnam's inability to clear the ball following Ali's downward header.
The teams looked destined to share the points, only for Adnan to give Van Lam no chance with his curling effort into the top corner.
Saudi Arabia had few problems starting their Group E campaign with a win as Juan Antonio Pizzi's side handed 10-man North Korea a 4-0 defeat in Dubai.
Hatan Bahbri put the three-times winners in front after 28 minutes when he cut inside from the left to hit a right-footed shot past Ri Myong-guk and Saudi Arabia doubled their lead nine minutes later when Mohammed Al Fatil flicked home Hussain Al Mogahwi's free kick.
Han Kwang Song was sent off for a second booking a minute before the interval while Salem Al Dawsari put the result beyond doubt 20 minutes from fulltime and Fahad Al Muwallad added a fourth in the 87th minute.