The second-half introduction of the 18-year-old spurred Falkirk to life and two assists in the space of five minutes shortly after the break ultimately earned the SPFL Championship side the bragging rights.
However, after a header from home captain Andy Geggan had buoyed the home side, Biabi took exception to an innocuous skirmish with the full-back Ross Millen 15 minutes from time, pursued him 30 yards up the pitch and attempted a head-butt.
The teenager looked in danger of completely losing the plot as another squabble erupted on the production of referee George Salmond's red card before he was escorted off the pitch by team-mate Ollie Durojaiye and Falkirk officials. The incident and the second-half goals betrayed what had gone before, with Dunfermline's wasteful finishing the biggest talking point of a largely tame first half.
Jonathan Page somehow skewed Ross Forbes' free kick into Falkirk's Jamie MacDonald from only two yards out and Michael Moffat glanced a Faissal El Bakhtaoui cross wide from merely six yards out.
Peter Houston, the Falkirk manager, gave his misfiring line-up until five minutes after the break to improve but the double change he then made helped inspire the visitors to victory. Biabi enlivened the Falkirk attack and, after getting the better of Page inside the box, his cutback was perfect for Taiwo to slot coolly in from the edge of the area.
Then, again. Biabi fed his team-mate 18 yards out and, this time, it was a deft chip with the outside of his right boot that speared Dunfermline's hopes of progressing.
From the comfort of a two-goal advantage, Falkirk almost contrived to throw away their quarter-final berth. The concession of Geggan's 69th-minute header from a Millen free-kick gave Dunfermline heart and Biabi's sending off ensured it was a long final quarter of an hour.