Mark Warburton was relieved to escape Palmerston with three points after admitting his Rangers side put in one of their worst displays of the season.
The Ibrox side looked frustrated as they struggled to break Queen of the South down.
But substitute Kenny Miller climbed off the bench to fire the winner within four minutes after his introduction on the hour mark.
The former Scotland striker's effort was jeered by the home support as it appeared to be soaring over the crossbar.
But it was the travelling support who had the last laugh as it suddenly fell towards goal, eventually nestling in the top corner for the 36-year-old's 100th Scottish league goal.
However, Warburton was unhappy his team failed to take confidence from the breakthrough and admitted Gers were well below standard as they clung on for the 1-0 victory which maintains their eight-point lead at the top of the Ladbrokes Championship.
The Englishman said: "It was an important three points but that's probably as poor as we've been all season in the last 20 minutes of the second half.
"We gave the ball away cheaply, sat too deep and gave them too much time and space. Wes Foderingham pulled off two or three outstanding saves. I've said before that we've played well and didn't get the rewards but today we'll take that because we played poorly for the last 20 minutes.
"It was a special strike from Kenny. He sets a good example. He came off the bench and got the goal which was really important.
"But as a squad can we be better? That was as poor as we've been all season in the last 20 minutes."
The Light Blues were without top scorer Martyn Waghorn for the first time since he was ruled out for eight weeks with a knee injury.
But Warburton denied the striker's lay-off was behind their sloppy showing.
"It's not about Martyn," he said. "I'm not going to say we won't miss a guy who has scored 28 goals - of course we will. But look at our bench - Kenny Miller, Gedion Zelalem, Nicky Law, Dean Shiels, Dom Ball, Harry Forrester - all really good players. We'll miss Martyn but it's about the squad."
Gers were grateful for goalkeeper Foderingham and the string of saves he pulled off to keep the Doonhamers at bay.
He denied Derek Lyle early in the first half before producing two great stops to keep out Ryan Conroy and Chris Higgins at the death.
And Warburton added: "I think he's an outstanding player. He's still young, but his reaction saves are first class.
"He's a baby in terms of being a keeper at 24. He has come from playing in front of five, six, 7000 to being at home in front of 45,000. He's learning from it, dealing with it and handling the pressure that comes with that."
Queen of the South boss James Fowler was disappointed his side could not capitalise from their late surge.
"I felt we were unfortunate not to take something from the game," he said.
"I felt we were camped in a wee bit in the first half, with Rangers dominating possession, but we tweaked it at half-time and that gave us more impetus.
"I felt in the second half we created enough chances to at least get a point."
Source: PA
