The Welshman came through a high-quality afternoon session 5-3 to lead 9-7 going into the evening's action in Sheffield.
Williams hit a break of 113 in the opening frame to lead 10-7 and after sharing the next two frames, a second big break of 113 to put him 12-8 in front.
World number 15 Carter had chances to claw his way back in frame 15, but Williams pushed on to keep his Crucible hopes alive.
The two-time world champion now faces Barry Hawkins in the last four, while the winner of the battle between John Higgins and Judd Trump playing Kyren Wilson.
After clinching his semi-final spot, Williams said: "It's going to be special and I will enjoy it more than any other semi-final I have been in.
"The tour is getting stronger, the standard is going up, I am getting older and I don't know how many more semi-final I will get.
"I am playing of the best stuff I have played for years. I am playing with a lot of confidence and feel relaxed."
