A draw in Tuesday's match in Amman would have been enough for the Blue Samurai to become the first team to qualify for the 2014 finals in Brazil but Yasuhito Endo missed a second half penalty as the Asian champions crashed to a 2-1 defeat.
According to Kyodo news agency, a laser trained on Endo when he was preparing to take the penalty while goalkeeper Eiji Kawashima was also targeted at the King Abdullah International Stadium.
"We will make a complaint," Kyodo quoted JFA chief Kuniya Daini as saying.
Endo said he was dazzled even before he stepped up for the penalty kick in the 71st minute.
"It wasn't just the penalty, there were other times in the game, too, but it (the laser beam) didn't really affect me," the 33-year-old midfielder told the news agency.
"I was focused and tried to place the ball where I wanted to put it, but the goalkeeper read it."
Japan coach Alberto Zaccheroni was also angry after two Jordan players made provocative gestures toward him, the report said.
"I'm not the kind of coach that likes to be provoked and threatened," Kyodo quoted him as saying.
Despite the defeat, Japan, on 13 points, are still six clear at the top of their group ahead of Jordan who leapfrogged Australia into second spot.
Japan host Australia at Saitama on June 4 and wrap up their campaign with a June 11 away match in Iraq.