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At 63-63, Robertson took the first frame with a respotted black but Carter took the second with a 94 break.
Carter won the third frame to go 2-1 up but Robertson fought back with a break of 43 and 57 to take a 3-2 lead.
However, Carter came from behind to seize victory, as he won three straight frames including an 84 break.
He will face off against Trump, who beat Ronnie O'Sullivan on Wednesday, in Friday's semi-final.
Semi-finals
Judd Trump v Ali Carter
Mark Selby/Marco Fu v Ding Junhui/Anthony Hamilton
Quarter-finals
Mark Selby v Marco Fu
Ding Junhui v Anthony Hamilton
Neil Robertson 3-5 Ali Carter
Ronnie O'Sullivan 3-5 Judd Trump
Round one
Stuart Bingham 1-5 Anthony Hamilton
Ronnie O'Sullivan 5-1 Liang Wenbo
Mark Selby 5-4 Ryan Day
Marco Fu 5-1 Anthony McGill
Barry Hawkins 0-5 Neil Robertson
Judd Trump 5-2 Mark King
Ding Junhui 5-4 John Higgins
Ali Carter 5-4 Shaun Murphy

After a fourth day of searches, 384 people have been rescued from the Annapurna trail, 216 of them foreigners, in the country's worst-ever trekking disaster.
Helicopters have been searching parts of the trail high as 5,790m (19,000ft) for survivors.
The government said the priority was to rescue 22 stranded hikers.
The trekkers are stranded at Thorung La pass, and face a shortage of food and water. Army helicopters tried to rescue them but failed after being unable to land.
The BBC's Andrew North in Nepal says that it is still unclear which climbers have been accounted for and which are still missing.
The task of the authorities is made more complicated because there are thousands of climbers in Nepal at this time of the year.
Army helicopters tried unsuccessfully to rescue them on Saturday between the Mustang and Dolpa districts, government rescue coordinator Suresh Acharya said.
Twenty bodies have been recovered but 19 bodies are still in the snow and a rescue team will be deployed on Sunday to dig them out and retrieve them by air, he said.
One of the main trekking routes - the Thorung La pass, the high point of the Annapurna Circuit - is now  "relatively safe" after the army cleared snow, he said earlier.
Rescuers have only limited resources and most of the missing and dead are believed to be at the maximum heights that helicopters can reach.
Nepalese, Japanese, Israeli, Canadian, Indian, Slovak, Vietnamese and Polish trekkers are said to be among the dead.
Many survivors have been left with severe frostbite and will have to have limbs amputated.
Nepal earns huge revenues from the tens of thousands of trekkers and climbers who flock to the Himalayas every year.
However, it remains a very poor country and the authorities here have struggled to cope with the scale of this disaster, our correspondent says.
This has been a particularly deadly year for Nepal's trekking and mountaineering industry.
An avalanche on Mount Everest in April killed 16 Sherpa guides and resulted in a significant drop in the number of expeditions to the world's highest peak.
The latest disaster comes at the height of the trekking period.
Nepal blizzards: Survival guide
Ali Carter beat Neil Robertson 5-3 in the Players Championship quarter-final in Llandudno, to set up a semi-final tie against Judd Trump on Friday.
[NEXT_CONCEPT]
Snowstorms and avalanches have killed 39 trekkers on a popular hiking route in the Himalayas, Nepali officials say.