Goals from Georgios Samaras and Scott Brown eased the Hoops to an impressive victory away to Caley in their fifth-round tie at the Tulloch Caledonian Stadium.
Both sides took time to come to terms with the windy conditions but the pendulum swung the visitors' way when Greece striker Samaras fired high past Jonny Tuffey in the 33rd minute to give them the interval lead.
In the 67th minute, after Caley defender David Proctor had conceded a penalty for a foul on Gary Hooper, Celtic skipper Brown slotted in the spot-kick for his third goal in three games to take the holders into Monday's draw at Hampden.
The result extended the Hoops' domestic winning run to 15 games and, being one point clear of Rangers at the top of the Scottish Premier League and in the final of the Scottish League Cup final, kept them on course for the domestic treble.
A first-half Eoin Doyle strike earned Hibernian a place in the last eight as they beat Kilmarnock 1-0 in another all-SPL tie.
The Irishman's goal was just reward for a fine display by the home side during an opening period which also saw an effort by James McPake ruled out and Tom Soares go close.
Paul Heffernan fired the visitors' best chance into the arms of the recalled Graham Stack.
Motherwell booked their place in the next round with an emphatic 6-0 victory over Morton at Fir Park.
Stuart McCall's men stunned their Scottish First Division opponents as they raced into a 5-0 half-time advantage.
Tom Hateley netted directly from a corner before Jamie Murphy, Shaun Hutchinson, Henrik Ojamaa and Nicky Law all netted for the hosts as the tie was all but decided before the interval.
Murphy then blasted home his second midway through the second half to add to Morton's woes and complete a one-sided scoreline.
Aberdeen had to come from behind to earn a replay as they drew 1-1 against Queen of the South.
Scott McLaughlin netted against the run of play after 54 minutes to give the Doonhamers hope of pulling off a shock against their top-flight rivals, who have fallen victim to a number of cup upsets in recent seasons.
However Scott Vernon scrambled the ball home after 67 minutes following a long Kari Arnason throw-in to keep alive the Dons' hopes of reaching the quarter-finals.
First Division outfit Ross County also forced a replay against SPL opposition as they drew 1-1 with St Mirren - but only after Richard Brittain missed a stoppage-time penalty.
The County captain had put his side ahead from the spot five minutes before half-time, only for the effort to be cancelled out within three minutes by a headed goal from St Mirren striker Steven Thompson.
Brittain had a golden opportunity to score a stoppage-time winner after Mark Corcoran was brought down by Lee Mair, but this time he sent his penalty sailing well over the bar.
The day's other scheduled fifth-round game - between Ayr United and Falkirk - was postponed.
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