Celtic eventually overcame some stodgy Raith Rovers defending and a churned-up pitch to book a place in the Scottish Cup quarter-finals.
A Charlie Mulgrew corner which clipped the post was the closest to a goal in a first-half starved of chances.
However, Kris Commons won and converted a penalty early in the second-half to break the deadlock.
And late goals from James Forrest and Charlie Mulgrew sealed the win for the Scottish Premier League side.
The Celtic defence looked nervous early on, with some edgy passing from Kelvin Wilson, but it was the visitors who created the first opportunity, as Mulgrew's deep corner was headed over by Victor Wanyama.
As the game moved out of its initial phase, Celtic were camped outside the Raith box but, perhaps in part because of a heavy, bumpy pitch, Neil Lennon's side could not create a clear shooting opportunity.
Raith finally mounted a purposeful counter-attack on the half-hour mark, but Brian Graham's header was weak and easily collected by goalkeeper Lukasz Zaluska.
Shortly after, Celtic were the width of the woodwork from scoring as Mulgrew's corner went through a clutch of players and deflected clear off the back post.
Graham could have caused a shock had he connected with a cross from the right at the started of the second half and Celtic responded with an Adam Matthews shot fizzing over the bar.
The second period was being played in a much more adventurous fashion and Tony Watt went close with a header that slipped just wide.
Graham forced an uncomfortable save from Zaluska with a 25-yard free-kick which dipped awkwardly in front of the Polish keeper.
But the visitors were handed a chance to break the deadlock when Commons went down under a Simon Mensing challenge just inside the box.
The Scotland international took the penalty himself and lashed the ball low to McGurn's left and into the corner of the net.
The goal sparked Raith into life and Pat Clarke passed up a chance to instantly level when he hooked a volley well over from a Jason Thomson cross.
Scott Brown should have doubled the visitors' lead when Gary Hooper's driven effort was parried into his path only six yards out, the Celtic captain blazing wide.
But fellow substitute Forrest settled any nerves Celtic might have had, going on a run down the inside-right channel and slotting the ball past McGurn.
Mulgrew piled on the misery for Raith late on with a delightful chip form the edge of the box which curled into the top corner.
Greig Spence's drive in the final moments flew wide and any hopes of a consolation for the Fifers were gone.