Queen of the South 1 Hibernian 2
The Dumfries men, playing in the quarter-finals for the first time in 30 years, stretched their SPL opponents to the limit.
Hibs manager John Collins admitted: "We did all the running in the opening half, but it took a 30-yard shot from David Murphy to give us the breakthrough.
"We were fortunate to get the lead back within minutes of Queens equalising, but their goal had given them spirit and they did have one or two chances of coming back into the game during the second half."
Queens boss Ian McCall praised his players for their performance and added: "We might just have pulled level and taken it to a replay in the second half.
"But I was a little unhappy at the way we lost that second goal which turned out to be the winner - we gave it away."
Queens had to back-pedal in the early stages as Hibs applied the pressure and Steven Fletcher was unlucky in the opening stages not to reach a well-placed pass from Scott Brown.
That early pressure was only relief for Queens by a run from Stephen Dobbie, but the Hibs defence dealt with it comfortably.
It took Hibs until the 18th minute to force the first corner of the match but the inswinger from Murphy was headed clear by the Dumfries defenders.
Willie Gibson eased the pressure with a couple of runs with shots from which he might have scored, but one went well over and the other was saved by the keeper.
Hibs came close to scoring when Michael Stewart had a shot from 25 yards only to see it deflected, leaving keeper Jamie MacDonald scrambling across his line and relieved to see it go outside the post.
As Hibs continued to press a high ball into the Queens area, it looked sure to provide an opening, but keeper MacDonald coolly took the ball off the head of Steven Fletcher.
Then on 45 minutes came the breakthrough that sent the 2,200 Hibs fans in the crowd to their feet. With no danger apparent Murphy latched on to the ball 35 yards out and took a couple of steps before unleashing a rocket which beat keeper MacDonald all the way as it screamed into his right-hand corner.
But Queens were level within a minute of the restart as Dobbie gained a throw out on the left.
It went to Jamie McQuilken who lofted it into the middle for John O'Neill to run in and head home an equaliser that caught keeper Simon Brown stranded in the middle of his goal.
Stung by this, Hibs swung into a quick response and were back in front four minutes later as Ivan Sproule crossed from the left for substitute Thomas Sowunmi to beat keeper MacDonald to the ball to head into the net.
Hibs had a let-off a few minutes later when a shot by O'Neill seemed goalbound until it was deflected by a defender for a corner.
Queens however came more into the game as they tried to pull themselves level and for a spell they were proving a real handful for the Hibs defence.
At one point Queens fans were left asking just how the ball didn't go into the Hibs net as a free-kick from Eric Paton was turned goalwards by Andrew Barrowman and it rolled tantalisingly along the line to the other side, where Jamie Adams could only side-foot it back and once again it ran along the line before being cleared by the rattled Hibs defenders.
Then in the dying minutes keeper Brown dived across his goal to pull down a shot from Neil Scally and put his side into the semi-finals.