Peter Pawlett scored the only goal on 40 minutes at New Bayview, with United top of pool B and safely into the last 16.
League 2 outfit Kelty Hearts moved into second spot on six points with one match remaining after a 3-1 triumph at Elgin City.
Creag Little put the hosts ahead after just five minutes at Borough Briggs, but Kevin Thomson’s men responded strongly in the second period. Cameron Russell equalised on 52 minutes, before Nathan Austin bagged a brace.
The Fifers will play Arbroath on Saturday, with the winner in with a chance of qualifying as one of the best second-placed teams.
St Mirren just need a point to top Group H after a 3-1 win at Stenhousemuir. Curtis Main, Kristian Dennis and Lee Erwin were on target for Jim Goodwin’s men, with Thomas Orr scoring for the Warriors. Elsewhere in that pool, Partick Thistle ran out 2-0 winners at home to Dumbarton, with Brian Graham scoring twice.
In Group A, Cove Rangers provided a shock result against Inverness, winning 3-1 at the Balmoral Stadium. Leighton McIntosh, Blair Yule and Mitch Megginson were on target, with Manny Duku bagging a late penalty for Billy Dodds’ men.
Brechin City surprised Alloa 1-0 at Glebe Park to dent the Wasps’ aspirations of progress, Queen’s Park won 2-1 at Annan, Clyde won on penalties at East Kilbride after a 2-2 draw and Kilmarnock defeated Morton on spot-kicks following a 1-1 draw at Rugby Park.
In Group E, Ayr United took pole position in the group with a 1-0 win away at Hamilton Accies.
In an all Championship affair, the game at the FOYS Stadium was settled by a 27th-minute strike by Cameron Salkeld.
Falkirk will have the opportunity to close the gap on the Honest Men to a single point on Wednesday night when they travel to Ainslie Park to face Edinburgh City.
Four other matches take place on Wednesday. Motherwell travel to Airdrieonians for a Lanarkshire derby at the Penny Cars Stadium, Brora Rangers host Ross County, Livingston take on Raith Rovers at the Tony Macaroni Arena and Dundee travel to Links Park to play Montrose.