Celtic had to come back from behind to seal their progress
Aiden McGeady returned from his club ban to help Celtic come from behind and earn a Scottish Cup win over Dundee.
Artur Boruc's fresh-air swipe in the 14th minute allowed Colin McMenamin to put the First Division side in front.
But Scott Brown levelled from close range after a mis-hit effort from Scott McDonald rebounded into his path.
McGeady, back for the first time since his dressing-room row with manager Gordon Strachan, curled in the winner on the stroke of half time.
With high winds and rain making life miserable, it was far from a vintage Celtic performance and there was a last-minute scare for the Scottish Premier League leaders when Boruc spilled a cross from McMenamin a yard from the line, but Glenn Loovens was on hand to clear.
It had been another blunder from the goalkeeper that handed the visitors a surprise lead early in the game.
The Poland international looked well placed to deal with David O'Brien's long pass forward, but he completely mis-timed his attempted clearance, allowing McMenamin to knock the ball into an empty net.
Celtic continued to stutter when striker McDonald drove wide of the target and Brown headed a Shunsuke Nakamura free kick into the side-netting.
McDonald had another shot that bounced off Jim Lauchlan's head before McGeady fired into the arms of former Parkhead keeper Rab Douglas from the rebound.
The home side's increasing pressure paid off when the energetic Brown drove down the left and into the Dundee box before pulling the ball back to McDonald.
A mistake from Boruc allowed McMenamin to roll in the opening goal
The Australian mis-hit his shot from 14 yards, but the ball ricocheted off several Dundee defenders to land in front of Brown, who stroked it home from close range.
With the interval looming, McGeady eased the home fans' nerves when he cut in from the left and curled a wonderful shot from the edge of the box high past Douglas.
Celtic were more relaxed after the break and McGeady had another effort from 20 yards tipped over the bar by Douglas.
Dundee were seldom seen as an attacking force, but McMenamin flashed a drive from 20 yards past the post on the hour mark to indicate a threat.
Douglas was required to pull off a fine save from Vennegoor of Hesselink's drive after the Dutchman had been put through by Nakamura's incisive pass.
McDonald then messed up another chance when he chipped over the crossbar from the edge of the box after neatly turning Lauchlan.
The Australian was left frustrated again when he created space from a Nakamura pass to curl in a shot from 12 yards only to see the ball fly inches wide.
Having missed several good opportunities to increase their lead, Celtic were relieved when Boruc's late mistake went unpunished.
Celtic manager Gordon Strachan: "We made enough chances and we're annoyed we didn't score more - our finishing was not at its best.
"What we didn't expect was the first goal, that was for sure, but the players sorted it out and got in front.
"We tried to get McGeady wider today because that's where he's most potent. But, if we want him out there, we must give him the ball.
"We're asking him to open up teams three or four times in a game and others can take advantage.
"There were a lot of things we were pleased with and the guys who came in needed a game."
Dundee manager Jocky Scott: "When you consider who we were playing and where we were playing, we've done well.
"The only criticism I have is that we gave the ball away far too often.
"But the work-rate and attitude was great and we can take a lot of positives from today and hopefully learn something from the way Celtic played as well."
Celtic: Boruc, Caddis, Loovens, McManus, Naylor, Nakamura, Scott Brown, Crosas (Robson 64), McGeady, Vennegoor of Hesselink (Samaras 70), McDonald.
Subs Not Used: Mark Brown, Caldwell, Mizuno.
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