Japan, who had previously taken 13 points from five group matches, missed the opportunity to become the first side to qualify for the 2014 finals (outside of hosts Brazil).
A point or better would have sealed a fifth consecutive appearance at the World Cup. Instead, Japan got nothing from a pulsating game at the King Abdullah Stadium.
Kagawa was reportedly heavily involved in Japan's attacking play, but his 69th-minute strike, a tidy finish applied to a bouncing ball eight yards from goal, came only after Jordan had gone 2-0 up thanks to strikes from Khalil Bani Ateyah and Ahmad Ibrahim.
Two minutes later, Yasuhito Endo had a golden chance to secure the point Japan required to qualify but missed from the penalty spot.