Khairul Amri then converted a penalty for Singapore but substitute Yousef Ahmad netted in stoppage time to help Jordan go through with a game to spare.
Singapore defender Baihakki Khaizan was sent off at the end of the first half and Jordan captain and goal-scorer Hayel was also shown the red card in the 70th minute.
Jordan are two points behind group leaders Oman. Syria are third on four points while Singapore have three.
"When Singapore scored the penalty it put my players under a lot of stress," coach Hossam Hassan told the Asian Football Confederation website (www.the-afc.com).
"But we kept our composure to score again and achieved what we wanted to do, qualify for the finals."
Twelve nations had already qualified for the 16-team finals - hosts Australia, holders Japan, North Korea, South Korea, Bahrain, Iran, Kuwait, Qatar, Saudi Arabia, United Arab Emirates, Oman and Uzbekistan.
The draw will be made in Sydney on March 26.
(Reporting by Sudipto Ganguly; editing by xx)
