The Englishman, 40, took the last two frames, sealing victory with a break of 55 to claim his first ranking title since the 2015 World Championship.
Bingham had led 4-0 in the early stages and came through a scrappy final session that saw a highest break of 63.
"Unbelievable," said the world number two. "To get my hands on another trophy means everything."
Compatriot Trump, 27, cut the early deficit to 5-3 by taking the last frame of the afternoon session and moved 7-6 and 8-7 ahead in the evening.
However, Bingham got back on level terms and, after Trump missed an early opportunity in the decider, it was the former world champion who prevailed with a clearance.
"I honestly felt that Judd outclassed me from the word go," said Bingham. "The first two frames were massive but it was only from his mistake that I cleared up and won.
"I've been knocking on the door since October, playing pretty well. I thought it wasn't going to happen here and hats off to Judd, from 4-0 down a lot of people would have crumbled and given up."
Trump said: "It was tough. I missed a few chances early on. I kind of threw it away in the first four frames.
"I missed too many easy balls and even tonight when I was getting back into it, I missed another easy ball. On the whole I did well to get back into it, it was just the odd shot here and there that cost me."
Stuart Bingham held his nerve in a tense final frame to beat Judd Trump 9-8 and win his first Welsh Open title.