Cilic will face either defending champion Roger Federer or unseeded South Korean Chung Hyeon, who play on Friday.
The unseeded Edmund, ranked 49, won 6-4, 3-6, 6-3, 6-4 in 2hr 49min on Rod Laver Arena and will face either 16-time Grand Slam champion Rafael Nadal or Marin Cilic for a place in the final.
Denis Shapovalov had lost the first two sets against Edmund in the fifth and deciding rubber when he blasted the ball away in frustration at 1-2 down in the third set.
Top seed Novak Djokovic brushed aside Kyle Edmund in straight sets to storm into quarters of 2016 US Open. In women's singles, second seed Angelique Kerber reached the last eight after defeating Petra Kvitova
Novak Djokovic's third round opponent Mikhail Youzhny retired in the first set while Rafael Nadal romped in to the last 16 with a straight sets win over Russian Andrey Kuznetsov
Andy Murray will play Croatia's Marin Cilic in the last four on Saturday in a rematch of the 2013 Queen's Club final
David Goffin came from two sets down to defeat British newcomer Kyle Edmund 3-6, 1-6, 6-2, 6-1, 6-0 and give Belgium a 1-0 lead in the Davis Cup final.