Kyrgios defeated the Italian, who beat Roger Federer in the last round, 5-7, 4-6, 6-3, 7-6 (7/5) 8-6 and will play the winner of Andy Murray and Grigor Dimitrov.
The Australian came back from 5-6 and a match point in the fourth set to force a tie-breaker which he won before winning the 5th set downing Seppi on Hisense Arena.
Afterwards Kyrgios said when he faced match point he thought back to his match with Frenchman Richard Gasquet at Wimbledon when he saved nine match points to win.
"It's crazy. I don't think it's sunk in yet," Kyrgios told reporters. "When I saw I had finally won the match it was incredible. It was the best feeling I ever had."
"They were unbelievable," Kyrgios said of the crowd. "I was definitely feeding off the crowd in the fourth, especially in the tiebreak it was massive.
"That was a momentum builder going into the fifth set."
Kyrgios is the first Australian to reach the quarter-finals of his home Grand Slam since Lleyton Hewitt in 2005.