The visitors were in control for large periods of the match but had to wait until the 78th minute for Chris Killen's winning goal.
Killen - a second-half substitute - stroked home his 16th international goal from the edge of the penalty area, curling his shot beyond the reach of substitute goalkeeper Abdullah Al Sudairy.
After chances had gone begging for the visitors they were relieved to see the 31-year-old's strike settle in the back of the net to extend their unbeaten run to nine games.
Chris Wood and Kosta Barbarouses had the best chances for the Kiwis in the first half, but both players wasted numerous opportunities.
When Wood looked certain to open the scoring he found Saudi goalkeeper Waleed Abdullah in fine form.
Abdullah doing brilliantly to get a fingertip to the Leicester City striker's shot to turn it just around the post.
The hosts looked dangerous on the break, but their finishing rarely brought a save out of Glen Moss in the New Zealand goal.
Barbarouses spurned another good opportunity early in the second half, however the earlier misses were all put behind them with 12 minutes remaining as Killen stroked them ahead.
The Chongqing striker received a pass from Barbarouses 25 yards from goal, controlled the ball instantly and curled his effort into the bottom corner.
Saudi went close immediately, but they wasted their best chance of the game as a tame effort was easily pouched by Moss.
