The match was a tense seesaw battle, with Carter establishing a 4-2 lead thanks to solid breaks and a 103. World number two Trump, however, roared back with a stunning 129 and three consecutive frames to take a 5-4 advantage.
But Carter, ranked number 10, remained composed, coolly sinking a tense 43 clearance in the 10th frame to level the score.
He then sealed his victory with a composed 64 break in the decider.
"You have to try and hold on to these special moments. To make a clearance like that in those circumstances and in front of a crowd like that is a feather in my cap," Carter told BBC Sport.
"For all the money I looked like I was going to go 5-3 in front and I lost my composure for a couple of frames.
Mark Allen and Mark Selby face off later today in a highly anticipated rematch of their 2023 World Championship semi-final.
The winner will take on Carter in the semi-finals on Saturday evening.
Trump, aiming to join snooker legends like Hendry and O'Sullivan in winning back-to-back Masters titles, was left to rue missed opportunities.
"He played well and I struggled," Trump said.
"He should never have really had that chance. It was terrible from me. I had two perfect chances to win the game and I'm obviously disappointed but he deserved to win."