Just 24 hours after Hibs fought back from 3-0 down to beat Falkirk 4-3 in the first semi-final, Hampden witnessed another seven-goal thriller with Anthony Stokes scoring the dramatic winner.
Kris Commons had taken just 82 seconds to break the deadlock when he gathered a pass from Victor Wanyama and turned 35 yards from goal before jinking past Mark Millar and dispatching a shot beyond the diving Radoslaw Cierzniak.
The Hoops should have doubled their advantage on 20 minutes when Emilio Izaguirre twisted past Willo Flood at the side of the box and chipped his cross on to the head of Mikael Lustig but he hit the inside of the post from four yards.
United made the most of that escape as Jon Daly pounced on a Kelvin Wilson slip to release Gary Mackay-Steven (24) and the winger exchanged passes with Ryan Gauld before placing his shot past Fraser Forster.
The Celtic defence was static six minutes later as Daly rose unchallenged to glance the ball past Forster from a Barry Douglas free-kick - 2-1.
Celtic responded immediately as Izaguirre, Charlie Mulgrew and Commons combined to open up United, and the left-back delivered another pinpoint chip for Wanyama to beat Cierzniak with a downward header.
The Hoops went 3-2 up when Efe Amrose burst forward and sent Izaguirre free on the left to once more cross into the six-yard box where the ball spun-off Commons (60) and over the line before the midfielder could react.
Cierzniak denied Commons a hat-trick when he deflected the ball round the post with his foot two minutes before Daly planted a header past Forster which went in off the upright from a brilliant Mackay-Steven cross - 3-3.
United scrambled away a last minute shot from James Forrest, who was also thwarted by the foot of Cierzniak in extra-time after exchanging passes with Anthony Stokes just a few minutes after Daly headed against the post at the other end.
Forrest then turned provider for the matchwinner when he crossed for Stokes (104) to net with a powerful header and book a place in the final against Hibernian next month.