Johnny Russell's first-half strike was enough to defeat Queen of the South and take Dundee United into the Scottish Communities League Cup quarter-finals.
Russell converted Michael Gardyne's low cross at the second attempt.
Second Division leaders Queens, unbeaten this season and having knocked Rangers out of the Ramsdens Cup, piled on the pressure in the second half.
But Russell's goal, United's first in four games, was enough to see the Scottish Premier League side through.
The visitors dominated the early stages of the match, but chances were at a premium and a weak Brian McLean shot, saved by Lee Robinson, was all they could show for their pressure.
United defender Gavin Gunning then had to look smart to clear a dangerous Willie Gibson after the winger had left Keith Watson stranded.
However, with a strong wind blowing in Dumfries, there was little to excite the crowd until Russell broke the deadlock.
Gardyne's low cross from the left was dummied by Jon Daly, but, while Robinson blocked Russell's initial effort, the Scotland Under-21 striker picked up the rebound and fired the ball into the roof of the net.
Moments later, Daly almost doubled United's lead, but his curling effort from 20 yards flew just wide.
Queens began to press for an equaliser and visiting goalkeeper Radoslav Cierzniak pushed a Nicky Clark header over the bar.
Paul Burns then headed over before the attacking midfielder set up Clark, whose effort was turned away by Cierzniak.
The home side continued to press but could not break through the United rearguard and Kevin Holt had to clear off the line after a United counter attack allowed Russell to round Robinson.
Queen of the South player-manager Allan Johnston: "We were playing against a top team and had them on the ropes there.
"In the second half, they couldn't get out of their own half.
"I'm just disappointed we couldn't get the goal.
"It was a great performance. Maybe the first half we showed them too much respect and stood off them a wee bit, or the conditions maybe didn't help, we gave them too much space in the first half.
"But the second half we got right on top of them, created a lot of problems and got a lot of crosses into the box, and were just unlucky not to get the goal."
Dundee United manager Peter Houston: "First half, we played some good stuff and created a few chances.
"We didn't do as well in the second half, but that is always to be expected, Queens have got to push men forward to try to get the goal.
"But I'm really pleased because, with the pedigree of the teams that Queens have beaten this season, coming down here is no mean task.
"What we were trying to do in the second half was to counter-attack and get in behind, but there were a few balls overhit.
"We didn't show the composure we should have done and that is disappointing, but the result in cup ties is always more important than the performance."