Inoha was on hand to side-foot in to an empty net after Japanese forward Shinji Kagawa had broken the offside trap and drawn a challenge from Qatar goalkeeper Qasem Burhan.
Japan fully deserved their victory in an entertaining match in which they were reduced to 10 men with an half an hour remaining after Maya Yoshida was sent off for a second yellow card for a foul on Qatar forward Yusef Ahmed.
"It was a very difficult match. Even after going a man down we had more of the possession. That a defender scored our final goal showed what Japanese football is all about," Japan coach Alberto Zaccheroni told reporters.
Qatar, ranked 105th by FIFA and bidding to reach the last four of the tournament for the first time, were not overawed by their opponents.
"We have shown the world that we are very strong and we showed that we can play good football, however we made mistakes and we were punished," Qatar coach Bruno Metsu said.
Qatar took a surprise lead in the 12th minute when their Uruguayan-born striker Sebastian Suria sprang the offside trap, sprinted down the right then cut inside Inoha and fired a left-foot shot under goalkeeper Eiji Kawashima.
Japan's Italian coach Zaccheroni was left complaining to officials that the referee should have blown for offside but television replays showed Inoha had played Soria onside.
The goal snapped Japan in to life after their sluggish start and they began dominating possession with the attacking trio of Kagawa, Shinji Okazaki and Keisuke Honda creating problems.
The equaliser came in the 28th minute when Okazaki snuk in to lift a shot over Burhan with Kagawa nodding the ball over the line from a metre out for his first goal of the tournament.
Japan continued to enjoy the majority of possession and better chances in the second half as their slick, sharp passing game left Qatar retreating deeper and leaving the dangerous Suria isolated in attack.
But the game changed in the 62nd minute when Yoshida was dismissed for bringing down Qatar's Ahmed.
Brazilian-born substitute Fabio Cesar whipped in the resulting free-kick which curled inside the near post, catching out Kawashima who was positioned behind the goal-line.
Qatar sniffed victory and pushed men forward but Japan regrouped and equalised when Kagawa neatly finished from inside the area in the 70th minute.
Both sides then spurned half chances but Inoha was in the right place at the right time to send the three-times winners into the last four.
"We had bravery to keep our attacking game," Zaccheroni said. "We lost one man because of sending off - players were looking tired - Qatar did not change their attacking play even though we were one man short so that gave us chance to come back."
