The full-back deliberately clawed away Paul Dixon's cross amid little danger to allow Craig Conway to stroke home the only goal.
United missed a penalty during Saturday's 1-0 defeat by third-placed Hearts but their third spot-kick in as many games moved them two points behind their European rivals.
Falkirk were unchanged following an encouraging draw at Hibernian but they rarely looked like extending their two-point lead over bottom club Inverness.
United manager Craig Levein made a tactical switch following Saturday's home loss, recalling midfielders Conway and David Robertson to move to a 4-5-1 formation.
But Francisco Sandaza led the line impressively and they had all the early possession and pressure, the most dangerous of which saw Lee Wilkie spark panic by heading Conway's free-kick across the six-yard box.
The home side then produced two sustained passing moves, which both ended with low crosses from Lee Bullen. Lukasz Zaluska smothered the first and Steve Lovell then dragged the right-back's cross just wide from 12 yards.
But United had regained the ascendancy before Scobbie literally handed them their 27th-minute opener. The Falkirk left-back inexplicably raised his hand above his head to stop Dixon's cross reaching Conway at the far side of the area.
The youngster could have blocked any forthcoming shot from Conway and only Dougie McDonald's whistle punctured a stunned silence around the Falkirk Stadium. Conway slid the penalty into the bottom corner after assuming responsibility from Francisco Sandaza, who missed against Hearts.
The goalscorer threatened twice more before the interval. He almost stole in to meet Mihael Kovacevic's cross but Dani Mallo punched clear and caught Sandaza's first-time lob.
Conway then forced Mallo to beat away his powerful effort after sharp interplay between Paul Caddis and Robertson had sent him clear in the right channel.
Falkirk brought on striker Mark Stewart for midfielder Arnau Riera as they switched to a 4-3-3 but United continued to look the more dangerous team.
Sandaza fired past the far post following a counter-attack and Wilkie headed wide from Conway's free-kick after Darren Barr had been booked for a cynical foul on Prince Buaben.
Steven Pressley followed him into the book for hacking down Conway after his sliced clearance had put Falkirk in trouble and Conway curled the resulting 25-yard free-kick wide.
Falkirk continued to cause their own problems. Pressley gave Burton O'Brien a pass he would not thank him for with his back to goal 30 yards out, and Caddis headed straight at Mallo from Conway's cross after the midfielder was dispossessed.
The home side forced the first save from Zaluska in the 66th minute after Bullen met O'Brien's corner from close range.
Falkirk had an even better chance moments later but Stewart headed just wide with Zaluska beaten after surging on to Bullen's lofted ball to the far post.
Higdon then headed on to the roof of the net after Stewart's flick-on but Falkirk could not break through the United defence in the final stages, and Darren Dods came close with a header at the other end.
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