Northern Ireland's Allen had one foot in the final at 16-14 up and on a break of 45 at Sheffield's Crucible Theatre in northern England.
But he missed a red into the middle, and Wu then responded with a back-to-the-wall contribution of 67 that cut the deficit to one frame at 15-16.
Following a re-rack in the next frame, a brilliant long pot from Wu paved the way for a break of 52, but he then missed a red with the rest.
Allen, however, could only manage six in reply before scattering the pack, but did not leave an easy pot on for Wu, who then went in off with the cue ball to let his opponent back in.
Both players then missed pots they would expect to make.
Allen, however, potted a brilliant long-range pink and seemed to have the match at his mercy when, in an ideal position, he missed the final black off its spot.
Wu, by far the youngest of the four semi-finalists at the age of 22, then knocked the black in to level at 16-16.
In the decider, he missed a difficult long red and that let Allen in for a break of 47 before the 40-year-old ran out of position after splitting the pack.
Neither man could make the decisive break, but when Allen left a tricky red on, Wu cut the ball into the bottom corner and cleaned up to reach his first world final, with the possibility that he could succeed Zhao Xintong, the 2025 winner, as a world champion from China.
Defeat left Allen, yet to reach a world final, still one tournament shy of completing snooker's Triple Crown after winning both the UK Championship and the Masters.