The Northern Irishman, ranked number four in the world, will face England's Shaun Murphy in the semi-finals.
Allen prevailed in a hard-fought match against Selby, which lasted a bumper four-and-a-half hours.
The Marks traded the first couple of frames, before Allen raced to a stunning 5-1 lead.
Selby would pull one back, but Allen had enough in the tank to seal his win over the Englishman before too long.
Speaking to the BBC after reaching the semi-finals, Allen said: "We both missed a few shots, but I can definitely say that is the most luck I have ever had in a match in a long time.
"It was ridiculous at times, there were two frames I really had no right to win."