Bingham told the BBC: "I was a bit disappointed in my own play, Dave gave me chances in the first three frames and I just took them.
"I got a bit of momentum going I'm just enjoying the moment out there and I'm feeling pretty relaxed."
Gilbert reached the last four of the World Championship last season and fell at the same stage this week.
He told the BBC: "I loved it even though I was playing poorly. I couldn't pot the tough balls and made it easy for Stuart.
"I've got nothing to worry about here, I had a bad day. People might say I'm a nearly man but I used to be the nowhere man so I'm making strides."
World number 17 Carter was only given an invite to the tournament after seven-time winner Ronnie O'Sullivan opted not to take part.
A break of 91 helped him to a 3-1 lead over 2015 champion Murphy and he also hit 70 and 97 to reach his third Triple Crown event final.
"When you get into the competition like I did you think you're on a free roll but when you get here it hits you in the face," Carter told BBC Sport.
"I now think it's mine to lose rather than just being happy to be here, that's my competitive edge.
"It's been a tough few years but I'm still here and still fighting.
"I've been to hell and back and it's special to be in the final but the job is not done yet."
Murphy made two centuries but after failing to build on a break of 56 in the sixth frame Carter went 4-2 ahead before going on to win.
"The way he finished that match off was quite special, it's not easy out there," Murphy told the BBC.
"It was very classily done and I wish him the best in the final."