There was never more than a frame’s gap between Williams and Perry in a nip-and-tuck affair as Perry rattled in the joint-highest break of the tournament, a 131, to force a decider.
But the Welshman produced a match-winning clearance of 75 to book his semi spot with a 5-4 win. And Higgins was made to work hard for victory after Carter fought back from 4-1 down to within a frame before Higgins wrapped up a 5-3 victory.
Carter, who had beaten Stephen Maguire 5-0 in the previous round, was kept in his seat for much of the opening two frames, Higgins following up a superb break of 103 with a 69 break in the second frame.
A surprisingly scrappy third frame followed, Higgins allowing Carter to pot his first ball after nearly 40 minutes of trying with some uncharacteristic errors as the man known as ‘The Captain’ gained lift off after a lengthy safety exchange.
A tough brown to the middle after an opening red let Higgins in the balls in the next frame, and he extended his advantage with a break of 57 to go into the mid-session interval 3-1 to the good.
Higgins got the better of a scrappy fifth frame to go one away from victory but he missed the final red with the frame at his mercy in the next and Carter cleared to keep in it.
The comeback continued in the next with two heavy-scoring visits but Higgins nipped the revival in the bud with two scoring visits of his own to book a semi final against Shaun Murphy.
An accomplished 72 break gave Perry the opener in his quarter-final and he was amongst the balls in the next frame until a missed red to the corner let Williams in to get on terms after a length safety exchange.
The Welshman proved the better in another frame-deciding safety exchange in the next, but a missed blue to the corner let Perry in for a frame winning 46 break to level at the interval.
A 45 break proved the catalyst for Perry to take a frame that included a weird one-handed safety shot from Williams, holding the tip of the cue with his left hand and prodding the ball rather than reaching around the table - but the Welshman responded with a 49 break to level at 3-3.
An outrageous fluke from Williams proved enough to put him a frame from victory, but Perry hit back superbly with a break of 131, the joint-highest of the tournament and the 147th of his career to take it the distance.
Perry sportingly called a foul on himself for a push shot in the critical decider and a fantastic opening red from Williams paved the way for a match-winning break of 75 and a semi final against Judd Trump.
Higgins will next play either Stuart Bingham or Shaun Murphy.