Group A winner Bingham, in his debut campaign in the competition surrendered a 2-0 advantage to fall 4-3 down, but claimed the final two frames to seal a remarkable turnaround 5-4 triumph over John Higgins.
Bingham made the better start and moved into a 2-0 lead after making a 45 break in the opening frame while he overcame a Higgins break of 36 in the second.
Higgins, aided by a 58 break, ended Bingham's 14-frame winning run in the third as the four-time world champion got on the scoreboard. He then levelled the match at two apiece courtesy of a break of 81.
Bingham moved back in front courtesy of a 132 break, but Higgins responded by taking the sixth frame 65-52, before coming from 50 points down to also win the seventh.
A break of 74 from Bingham levelled the match, taking it to a sudden-death decider. The 36-year-old maintained the momentum in the final frame, taking it and the match with a break of 57.
Concluding at 1.25am, the second semi-final was another close-fought affair and this time it was Trump who prevailed, beating Neil Robertson in a deciding frame.
At 3-1 down and needing a snooker in the fifth, things looked bleak for Trump, but he battled back before winning in the early hours. of the morning.
"It's unbelievable. To be in the final is unreal," Bingham told Sky Sports. "I'm just delighted. There was pressure. I felt nervous the whole way through.
"Playing John, without the rankings, is the best player in the world. To come through the match is great. I think everyone is happy it's not a Higgins and Judd final, as that would be boring."