Robertson beat fellow former world champion Murphy 5-4 while Carter came from two frames down to triumph 5-2 over Michael Holt.
The other semi-final on Saturday will be the in-form Barry Hawkins versus Marco Fu as they won 5-1 against world number one Mark Selby and 5-3 against Matthew Stevens respectively, with Fu bagging the only century of the session - a 126 - not compiled by Murphy, who had a pair of them.
Robertson opened a match featuring plenty of excellent safety play with an assured break in the seventies to ease to the first frame, Murphy gifting him the chance with a misjudged safety when rolling the cue ball softly into the pack off the right cushion.
Murphy found some rhythm in the second frame to level after a wicked kick on the pink for Robertson with the table wide open, and the Rotherham cueman moved ahead in a scrappy third frame containing double-kisses, in-offs and unintended pots.
The 2005 world champion took a chance in opening up the reds in frame four. It did not pay off as he missed his pot in doing so – and Robertson took full advantage with an 88 break to see them all square at the interval.
The Australian star – who had also won their previous five meetings – controlled the frame following their return despite missing a relatively easy frame ball black. Murphy’s hopes of a comeback in the frame were sunk when he just caught the black when coming off the bottom cushion while snookered on a red –eagle-eyed referee Jan Verhaas got the decision spot-on.
Robertson was soon a frame from victory as an out-of-sorts Murphy suffered bad luck alongside his wayward potting. At 4-2, no one would have bet on a Murphy win.
However he is at his best when his back is to the wall – and he certainly came out fighting here. After slamming down a brilliant long red, he expertly opened up the pack two balls later and capitalised fully on the chance, making 107. It could and perhaps should have been a 142, but he contrived to miss the final short red.
He was not finished: with 2010 world champion Robertson again stuck to his seat, Murphy tiptoed among the reds before flexing his cue in putting down a powerful doubled red to the middle pocket on his way to a 133 total clearance.
All the momentum was with him heading into the decider, but it was not to be. Robertson was unlucky to miss an early red but potted the next long chance before Murphy could build a lead of any note. It was his first pot for almost half an hour.
He reached the thirties then chances came and went for both players before Murphy left a small gap for the Thunder from Down Under to sneak through and pot a red. He cleared to the blue to finish the night’s snooker seconds after a brow-beaten Selby conceded in his match against Hawkins.
Marco Fu 5-3 Matthew Stevens
Allister Carter 5-2 Michael Holt
Neil Robertson 5-4 Shaun Murphy
Barry Hawkins 5-1 Mark Selby
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