Trump scored breaks of 108 and 129 to force a deciding frame, but Bingham held his nerve to take the victory by 17 frames to 16.
Bingham, 38, becomes the oldest first-time Crucible finalist since Ray Reardon took the title in 1978.
"I want to see my mum and dad and my wife to share it with them. It's unbelievable the way I got over that line. To come out at 16-all, the adrenaline and atmosphere was brilliant," Bingham told Sky Sports News.
"The emotions are flowing, I can't believe it. I've had a few tears already, and I'm trying to hold myself together."