Another substitute, New Zealand's Lucas Imrie also had a late chance to steal all three points but his header from a Moses Dyer-ball was a tough chance and he could not finish.
The draw sees New Zealand and Vietnam level on one point in Group E behind favourites France who are top after they defeated Honduras 3-0 to open in style.
New Zealand were put under early pressure by the FIFA U-20 World Cup debutants who showed little signs of nerves on the big stage.
Forwards Thanh Binh Dinh and Duc Chinh Ha were a constant threat for Vietnam going forward and could have managed the opening goal but for some good work in goal from Michael Woud who had a busy first half.
Dinh had the best scoring chance early on when he found space in the box but he dragged it wide. Two minutes later, Ha could not convert from close range as Vietnam had their second scoring chance in as many minutes.
Van Hau Doan got in on the act with a long range shot which Woud tipped over the bar with the faintest of touches to keep a clean sheet heading into half time.
New Zealand finished the half as the stronger team and the best chance so far fell to Noah Billingsley from a Clayton Lewis corner, but he could not get his header on target.
New Zealand carried that momentum in the second half when forward Bevan and captain Lewis had half chances they could not convert. Cameron earned a free kick just outside the box but the All White's shot was always over the bar.
In the second half, Vietnam looked dangerous on the counter-attack, but the last 45 minutes was dominated by New Zealand who found their rhythm. They walked off the pitch disappointed they didn't get more from the game.
Bazeley made three changes with Rogerson, Lucas Imrie and James McGarry coming into the fold, but for all of the chances the men in white created they could not combine to break down a red wall of defence.
New Zealand are back in action on Thursday night (11pm NZT) where they meet Honduras in what is a vital match in their chances of making the Round of 16.
New Zealand 0 Vietnam 0 HT 0-0
NZ: Michael Woud, 2. Dane Ingham, 3. Sean Liddicoat, 5. Hunter Ashworth, 8. Moses Dyer, 9. Noah Billingsley (20. Lucas Imrie 80') 10. Clayton Lewis ©, 11. Henry Cameron (17. Logan Rogerson 69'), 14. Jack-Henry Sinclair (13. James McGarry), 15. Reese Cox, 19. Myer Bevan
Subs not used: 4. Luke Johnson, 6. Joe Bell, 7. Connor Probert, 12. Cameron Brown, 16. Callum McCowatt, 18. Sarpreet Singh, 21 Conor Tracey.