Carter led 5-3 from the first session and two scintillating frames saw each player take one each.
Carter's cautious but clinical approach paid off when he edged a defensive 11th frame to lead 7-4, before an unfortunate foul by Murphy allowed the former world number two to move to within a frame of victory at 8-4.
But Murphy emerged after the mid-session interval a new man and played some brilliantly courageous snooker to erode Carter's lead frame by frame. Murphy struck a 105 break to move to within a frame of Carter.
Murphy forced a decider and, when the beleaguered Carter missed a straight red with the rest, wrapped up the match in awesome fashion to complete a classic comeback and a final against Mark Davis or Mark Selby.
Carter extended his lead in the first frame of the evening session, pouncing on some misfortune for Murphy, whose attempt to break the cluster of reds saw the cue ball nestle among them. Carter surpassed Murphy’s 46 to take the first frame in clinical fashion at the Barbican Centre in York.
Carter appeared on course to make it 7-3 but a loose red by the Essex man let Murphy back in. After the two were locked in a safety tussle, Murphy showed character to pot a vital red that, had he missed, Carter would have wrapped up the frame. But Murphy kept his head to reduce his opponent's lead to 6-4.
The next frame was cat and mouse for long periods, with both players exchanging safety shots until Carter rifled in a red left by Murphy before building another frame-winning break that ensured he re-opened his three-frame advantage at 7-4.
Carter was in full flow at this point and raced into a 60-0 lead, but was pegged back when he overcooked a red that forced him to play safe. Murphy opened the pack of reds with a curious shot, but Carter elected to leave a straightforward red open, which his counterpart duly potted – only for his misfortune to strike again as the cue ball rolled straight into the middle pocket.
Carter survived another bout of safety to move to within a frame of victory before the mid-session interval. But the fight back was about to begin.
Murphy struck back after the break with some brilliantly gutsy snooker. He showed bravery and determination to attack every ball, and the tactics served him well as he produced three thrilling frames to erode Carter’s lead.
Murphy surrendered a whopping 59 points in fouls, but what he lacked in fortune in replaced with mettle, pouncing on a loose red by Carter to slam another long red in on his way to a break of 105 and reduce his lead to 8-7.
Carter still only needed one frame to win it, but his confidence was ebbing away in the face of a ruthless Murphy onslaught. Murphy was in imperious form as he played without fear, winning the next frame in thrilling style to level the match at 8-8.
The deciding frame was a predictably cagey affair until Murphy sunk a brilliant red into the middle pocket that put him back in control - and was typical of the kind of snooker that got Murphy back into the game after the mid-session interval. Murphy kept his nerve to beat a broken Carter, and reach the final of the tournament he won in 2008.
First session report
Carter, who beat Stuart Bingham en route to making his first semi-final of the season, was never behind against the 2008 champion and two outrageous frame-winning flukes from the man they call ‘The Captain’ proved the only difference in a match-up that always looked close on paper.
Murphy, who edged out teenage Belgian Luca Brecel 6-5 to make the semi, was off and running in the opener with a lovely long red after Carter missed a red to the left corner. But with a 42-point lead and the frame seemingly at his mercy, a couple of uncharacteristic missed pots and an unfortunate in-off from the yellow let Carter off the hook and he took full advantage to take it on the pink.
Following a re-racked start to the second, Murphy was first in again but another missed red let Carter in for a frame winning 77 break. Murphy was yet again in first in the third and looked to be cruising before running slightly out of position with his break on 42.
There followed a moment of controversy when the referee warned Murphy that he must hit a red after two successive escape misses, although it appeared there was no red that he could see both sides of.
Justice may have been done, however, as Carter ran out of position on his final red with the frame in his pocket and Murphy cleared to the pink to pull one back.
Buoyed by pinching the last, Murphy kept Carter in his chair after his break off, knocking in a superb 89 to draw level at the interval. For once, Carter was the first man in the balls on resumption, two scoring visits enough to restore his frame advantage as Murphy failed to find the snookers he required.
An outrageous fluke saw Carter steal the next after Murphy had established an early lead before running out of position. Forced into a tough pot to the middle, the red flew off the jaw of its intended middle pocket target and into the opposite corner. It left Carter in prime position to knock in a superb 101 clearance to re-establish his two frame lead.
Murphy vented his frustration with a superb attacking 34 break at the start of the seventh before a fabulous long red set up a snooker from which he reduced the deficit to just one.
But with a tight final frame of the session drifting over the 40 minute mark, another outrageous fluke from Carter set up a frame winning chance. And although Carter missed the double on the one red he required to make frame safe, Murphy’s pot attempt jumped out of the jaws and off the table to leave him two behind.