Late goals from John Hartson and Paul Lambert killed off brave Ross County and ensured their was no Scottish Cup upset at Celtic Park.
John Hartson headed home Celtic's first goal
The Dingwall side frustrated an out of sorts Celtic in the first half with an impressive defensive display.
And after the break David Winters and Graham Bayne went close to scoring for the First Division underdogs.
Hartson stooped to head home after 74 minutes and Lambert's deflected shot sailed into the net four minutes later.
Celtic manager Martin O'Neill made three changes to the team which mauled Rangers at Parkhead last week, with Paul Lambert, Ross Wallace and Shaun Maloney coming in for Stanislav Varga, Stilian Petrov and Alan Thompson.
The Premier League leaders looked sluggish inthe early stages and found it difficult to break down the stubborn First Division side, who were packing men behind the ball.
County gave the subdued home support a fright after 14 minutes when the ball broke to Mark McCulloch and he fired just past the angle from 25 yards.
Hartson had the ball in the net in the 24th minute when he muscled Sean Webb off the ball, from Henrik Larsson's flick over the top, but referee Iain Brines deemed the challenge illegal.
Celtic carved out a great chance in the 27th minute when Neil Lennon picked out Shaun Maloney on the edge of the penalty box but his shot was well kept out by the diving Graeme Smith.
There was almost some embarrassment for his County's Webb in the 34th minute when he slid in to cut out Wallace's hopeful punt downfield and sliced the ball narrowly wide of his own goal for a corner.
Celtic looked more purposeful immediately after the break and Wallace and Maloney combined to give Larsson an opening, but his shot was too weak to trouble Smith.
But it was County who created the best chances and a ragged Celtic side were hanging on.
In the 56th minute John Rankin fired the ball across goal and Robert Douglas spilled the ball straight to David Winters.
The striker picked up the ball and hit a powerful shot which the Scotland goalkeeper blocked well to save his own blushes.
Winters was presented with another opening but he chipped just wide before Sutton blazed wide at the other end from 20 yards.
The visitors threatened again just after the hour when Jim Lauchlan got on the end of Rankin's corner but was unable to direct his header on target.
Stephen Pearson made his Celtic debut
County went even closer to grabbing the lead, four minutes later, when Rankin picked out Graham Bayne unmarked in the box but he headed inches over the bar.
With 16 minutes remaining, Hartson found room on the edge of the six yard box after a appearing to shove Lauchlan to the ground and the Welsh striker glanced home a cross from Wallace.
Larsson could have sealed the result twice when first his strike was saved by the feet of Smith before heading against the post.
But Lambert did put the game out of the visitors' reach in the 78th minute when his long-range strike took a deflection before nestling in the top corner of the net.
Stephen Pearson came on to make his Celtic debut in the 80th minute and generated the biggest cheer of the day.
Celtic: Douglas, Balde, Sutton, McNamara, Agathe (Pearson 80), Lambert, Maloney (Petrov 66), Lennon, Wallace (Beattie 80), Larsson, Hartson. Subs Not Used: Hedman, Varga.
Ross County: Smith, McCunnie, McCulloch, Lauchlan, Robertson, Webb, Hannah, Rankin, O'Donnell (Higgins 82), Winters (Gethins 76), Bayne (Cowie 81). Subs Not Used: Fridge, McGarry. Booked: Lauchlan.
Referee: I Brines