Trailing to a Derek Riordan penalty, Jim McIntyre equalised when he turned in a cross from Barry Robson after 73 minutes before substitute Jason Scotland struck the winner.
But a second-half booking for McIntyre after a flare-up with Hibs goalkeeper Simon Brown means he will miss the final.
Hibs threatened early on and Tony Bullock did well to get his hands to a Stephen Glass shot after just 45 seconds.
Riordan was next to try his luck with a 20-yad chip which drifted just over the top.
It took United some time to get into their stride and it was not until the 23rd minute that Grant Brebner had their first effort on goal but his wayward volley did not trouble Simon Brown.
Stevie Crawford then curled a 20-yard shot just a yard too high before United made claims for a penalty when a cross from Barry Robson appeared to strike the arm of Steven Whittaker, but referee Mike McCurry waved play on.
Hibs finished the first half strongly as United looked uncertain at the back. A mistake by Paul Ritchie gifted a chance to Steven Fletcher but he hooked the ball over the top from only four yards.
Garry O'Connor then sent a header directly at Bullock from just ten yards as another chance was missed.
Scott Brown tried to break the deadlock before half-time with a long-range effort which flashed wide.
Riordan shot Hibs ahead from the penalty spot after 58 minutes following Robson's foul on Dean Shiels.
But United hit back. McIntyre stunned the Hibs support with an equaliser and then, with 14 minutes left, Scotland took a pass from Mark Kerr and drilled a low shot past Simon Brown from the edge of the area.