Aniket Jadhav struck early for India and Aman Chetri doubled their advantage only for Sulaiman Al Dhuwayhi to restore hope for the Saudis 11 minutes before the interval.
Saudi Arabia's Mali missed from the spot with the last kick of the first half. But Feras Al Brikan's quick-fire brace in the final 10 minutes looked to have won it for the Saudis before captain Wangjam kept India in the tournament at the death.
The hosts will now need to defeat Iran in their final Group A outing on Wednesday to stand any chance of reaching the quarter-finals while the Saudis face the same predicament against the United Arab Emirates.
After the match, India head coach Nicolai Adam said: "It's really difficult for me to say anything after this game - I'm extremely proud of the boys. The support tonight was spectacular. People will now go to work tomorrow and talk about this game, as will people who watched on TV. We have created something.
"Saudi Arabia are a good team with short moves and good passing. With all respect to UAE, I think the Saudi team is two times better. I said to the boys before the game, awe win or we learn'. I told them if we believe in it then it's possible. I believed and the boys believed and finally it worked. We had a much better balance between offence and defence today."
