Banks saved two spot-kicks to decide the shoot-out, having earlier produced some excellent stops to keep the SPL League side on level terms.
County made a lively start to the game and Scott Morrison drove the ball into the United box but the flag went up as Martin Scott ghosted in at the back post.
At the other end David Goodwillie threatened on the edge of the 18 yard-box but Stuart Kettlewell slid in to scoop the ball clear before the United striker could pull the trigger.
But County created another chance in the seventh minute when Corcoran won the ball from United defender Sean Dillon but his ball across the six yard box was just missed by the in-rushing Andy Barrowman.
United were forced to replace Paul Dixon with Barry Douglas in the 27th minute.
Goodwillie was causing the County defence problems and it took a well-timed challenge by Richard Brittain stopped him in his tracks.
For the second match in succession Jason Marr was forced to retire early after he took a nasty head knock from the boot of Goodwillie. The on-loan Celtic player was replaced by Gary Miller.
Andy Barrowman burst through the middle of the United defence just before the break but with options on either side he went on his own but his shot was well wide of target.
County were almost in on the stroke of half time when Scott threaded the ball through to Steven Craig but he was upended on the edge of the box. Corcoran took the free kick but his ball into the box was easily cleared.
The home side forced a corner at the start of the second half and United centre back Gary Kenneth was headed over his own crossbar to avert a goal scoring opportunity.
United pushed for the opener and County goalkeeper Michael McGovern stopped headers from Jonathan Daly and Goodwillie.
Barrowman came close to breaking the deadlock in the 58th minute when he met a Brittain corner kick but the ball was knocked off the line by team-mate Craig.
But it was the home side who almost took the lead in the 58th minute when Scott's diagonal ball sent Barrowman clear but United keeper Steven Banks raced from his line to block his shot.
County were enjoying a good spell of pressure but in a breakaway move David Robertson fired a ball across the County goal but fortunately for the home side there were no takers.
But the first division side were on the ascendancy and Morrison let fly with a 25 yard shot which Banks managed to scramble away.
Brittain was next up with a blistering shot which Banks somehow managed to push onto the bar, and a minute later the United keeper produced heroics again with another wonder save from Scott which again deflected onto the bar.
County had the first chance of extra time but Corcoran's cross was just too high for Barrowman.
The home side had a chance with seconds to go when substitute Iain Vigurs raced down the left and his cut back was swept over the crossbar by Scott.
Chances were few and far between in extra-time and it was down to penalties.
United won 4-3 with Goodwillie scoring the vital kick after misses from County's Iain Vigurs and Scott Morrison.