Celtic kept on course for a domestic treble by scoring four second-half goals in a 4-0 win over ten-man Dundee United at Tannadice to book a place in the Scottish Cup semi-finals.
United had Robbie Neilson sent-off in the first half, before Joe Ledley, Georgios Samaras, Anthony Stokes and Scott Brown netted for the holders, who join Hibernian in the draw for the last four.
The winner of the quarter-final tie between Motherwell and Aberdeen, as well as the winner of a replay between St Mirren and Hearts, will complete the semi-final line-up.
"It was such a difficult game today. I didn't think we played anywhere near our best first half and that's credit to Dundee United," Celtic manager Neil Lennon said.
"They made life difficult for us and I thought they were brighter and first to the ball. We really tore into them at half-time but in the second-half we were magnificent."
Meanwhile, Lennon's United counterpart was proud of his players despite the margin of defeat.
"I'm really disappointed. I thought we dominated the first half in many ways," Peter Houston said.
"It's hard enough to beat the champions elect with 11 players never mind 10. But for long spells of the game I was exceptionally proud of the players as they put in a lot of effort."
United, who had knocked out Celtic's Old Firm rivals Rangers in the previous round, started brightly and Jon Daly came within inches of giving the home side the lead in the eighth minute when he bulleted a header just wide from Willo Flood's cross.
Johnny Russell should have done better when he met Gary Mackay-Steven's cut-back 10 yards out but the striker failed to catch his shot properly and Fraser Forster was able to make the save.
However, the home side's good work was undone in the 28th minute when Neilson was shown a straight red for leading with his elbow in an aerial challenge with Samaras.
United brought on Garry Kenneth for Mackay-Steven to shore up the defence and the substitute's first task was to block a shot from Ledley from a Stokes cross.
The game remained goalless at the interval, but Celtic started the second-half on the front foot and made the breakthrough in the 53rd minute through Ledley.
Good work from Stokes down the left saw him put a dangerous ball into the box and Welsh international Ledley beat Gavin Gunning to the ball to blast it into the net from close range.
United goalkeeper Dusan Pernis was equal to Stokes' drilled shot from 20 yards before the hosts missed a great chance to level matters midway through the second half.
Milos Lacny was sent clear on goal by Russell's headed flick on but the substitute dragged his shot wide of the target with only the 'keeper to beat.
And United were left to rue that missed opportunity when Celtic doubled their advantage moments later.
Charlie Mulgrew's delivery from a free-kick wide on the left was excellent and Samaras rose unmarked to bullet a header past the helpless Pernis.
Celtic had the chance to put the game out of sight in the 79th minute when James Forrest used his pace to breach the United defence but Stokes sent his shot from the cut-back crashing off the bar from 10 yards with the goal gaping.
However the Irish striker got a goal with four minutes remaining to put the icing on the cake for Celtic as he beat the offside trap to fire an angled shot past Pernis from Ledley's pass.
Things got worse for United in the final minute when Brown slotted home from the penalty spot after Samaras had been chopped down by Sean Dillon in the box.