Dundee United set up a Scottish Cup fifth-round visit to Hamilton with an easy win over East Stirlingshire.
Prince Buaben drilled the visitors in front, while Darren Dodds and Jon Daly added headers before the interval.
Johnny Russell, making his first start, converted a penalty on 55 minutes after his enterprising run was brought to a crude halt by Craig Tully.
With a strong wind at their backs in the first half, Shire caused United a few problems before capitulating.
Home manager and former United hero Jim McInally handed a debut to 17-year-old goalkeeper Jamie Newman after Falkirk recalled Jamie Barclay from his loan.
And despite shipping four goals, the youngster can be satisfied with his performance against Premier League opponents.
The Third Division side had a glorious chance to take a fifth-minute lead when Craig Donaldson played Colin Cramb clean through as United's offside trap failed, but the veteran forward chipped over the bar.
Daly was off target with his side-foot strike from 12 yards, and teenage forward Russell failed to get a strong enough touch to steer the ball past Newman, who had strayed off his line.
Buaben shot just over from 20 yards after a positive run by Russell, but the midfielder found the back of the net after 15th minutes.
Following a neat passing move down the left, Buaben fired a daisy-cutter into the bottom corner from 20 yards.
Shire threatened again, with Andy Rodgers narrowly missing the target from the edge of their box and Brian Graham heading into the side-netting after David Dunn's long-range effort had deflected wide.
Michael Bolochoweckyj then shot just wide of Lukasz Zaluska's near post as the United defence hesitated following a long throw.
Daly headed over the crossbar from Russell's brilliant cross before Dods nodded home from close range on 40 minutes after Paul Dixon had curled a corner into the six-yard box.
Daly added a powerful header from 10 yards two minutes later following an inviting cross by Sean Dillon, while Newman blocked Willo Flood's low drive.
Ten minutes into the second half, the lively Russell twisted his way into the penalty box only to be upended by Tully.
He picked himself up and slipped the spot-kick past Newman, with the diving, rookie keeper unlucky not to get a touch on the ball.
The game petered out in the last half hour but David Robertson went close to extending the visitors' lead when he clipped a shot against the outside of the post in the closing stages.
Shire substitute Mark McKenzie raced clear of the United defence late on but his shot was saved by Zaluska, while Newman produced a fine stop to deny Daly.
East Stirlingshire manager Jim McInally:
"When we had the wind behind us in the first half, we had to be a bit better.
"We missed a great early chance and you just can't do that. After they scored we had a good spell but then they picked us off too easily.
"The second half was a waste of time, in terms of the conditions but the players kept at it.
"I'm delighted for our young goalkeeper because he had a hard day and that will stand him in good stead for the rest of his career."
Dundee United manager Craig Levein:
"This was always going to be a difficult game. On paper it looks easy and everyone expects us to win but we didn't slip up and we were very professional.
"All credit to the lads for getting us through to the next round.
"I felt we let the tempo drop when we got in front as if that was enough but 1-0 is never enough so I really was having a go at them because I felt we eased off after a fantastic start.
"But we picked it up again and it was no coincidence that when we did we got a second and third goal.
"And we were very professional in the second half. We passed the ball well and created opportunities."
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