Match ends, Rangers 2, Celtic 0.
Rangers ended Celtic's four-year Scottish Cup reign - and condemned them to a first trophyless season since 2010 - to reach the quarter-finals with Old Firm derby victory at Ibrox.
Steven Davis' overhead kick and a Jonjoe Kenny own goal before half-time decided a gripping, chance-laden game.
Celtic's wasteful finishing culminated in Odsonne Edouard's weak penalty being saved in the closing stages.
Steven Gerrard's side host St Johnstone in the last eight next weekend.
Having crushed Celtic's bid for a 10th consecutive league title, Rangers are now three wins from adding the Scottish Cup after extending their 100% home record in domestic football this season.
They showed a ruthless edge the visitors couldn't match as the Ibrox men made it five Old Firm games unbeaten, including four wins.
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- Reaction & as it happened
Five years and a day since their last Scottish Cup defeat, a shootout loss to Rangers, Celtic again fell to their biggest rivals. From the quadruple treble to empty handed in the space of four months.
Celtic had their moments. Plenty of them. Stephen Welsh missed a glaring opportunity for an equaliser and Edouard and Mohamed Elyounoussi were the main culprits after the break.
Ultimately, their lack of ruthlessness was clinically exposed by a Rangers side who gobbled up the two first-half chances that fell their way.
Joe Aribo, the game's stand-out performer, was the architect of both, tormenting left-back Diego Laxalt who was put out of his misery when substituted at half-time.
By then, the damage was done and would prove insurmountable.
The opening scorer - Davis - was as unlikely as the goal's execution. Aribo took possession wide right when referee Bobby Madden played advantage after Ryan Kent was clattered by Welsh 30 yards out.
The midfielder jinked inside and tried a shot that rebounded off Kristoffer Ajer and looped into the air where 36-year-old Davis applied an acrobatic and emphatic finish for just his second goal of the season.
When Celtic roused themselves to respond, they ought to have been level. Edouard's stinging shot drew a save from Allan McGregor before Ryan Christie dragged wide and Callum McGregor had a drive deflected narrowly over.
John Kennedy's side kept up the pressure and young centre-back Welsh somehow fired away from goal at point-blank range from David Turnbull's delivery.
Rangers hit back, with Nathan Patterson's raids down the right flank an impactful feature of their play. The 19-year-old full-back darted past Laxalt and Scott Brown before lashing a shot into the side-netting.
The second goal come from the same area as Rangers preyed on weak link Laxalt. Aribo skinned the Uruguayan far too easily and his cutback was turned home by Kenny under pressure from Kent.
Celtic - with Greg Taylor on for Laxalt at the break - had it all to do but resumed with intent as Turnbull's 20-strike brought an impressive diving stop from McGregor.
Even better was to come from the Rangers goalkeeper, getting his body in the way of an Elyounoussi shot after the Norwegian danced past Filip Helander and Connor Goldson. The rebound fell to Edouard, who flicked the ball too high from close range.
Celtic's finishing was again their downfall when Elyounoussi took a heavy touch that allowed McGregor to gather after Edouard's square pass left the goal gaping.
Rangers attempted to kill off the tie on the counter-attack, but Alfredo Morelos could only poke a shot at Scott Bain after the Celtic goalkeeper couldn't hold a thumping Kent effort.
The hosts' lead looked set to be halved 11 minutes from time when Aribo pushed substitute Leigh Griffiths to concede a penalty. Edouard's effort lacked power, though, as McGregor dived to his right to save.
The final act was Griffiths lashing over from a corner in injury time, summing up Celtic's day as Rangers dream of a first league and cup double since 2011.
Man of the match - Joe Aribo
What did we learn?
Gerrard's Rangers have perfected the knack of beating Celtic without operating at their best.
The early lead gave them something to protect and they lived dangerously at times to weather the storm of Celtic's early second-half pressure and complete the task.
Rangers have arguably been outperformed in the last three Old Firm games, but they have the mentality and talent to grind out a positive result.
For Celtic, this defeat ends any chance interim manager Kennedy had of staying in the role beyond summer. The club's miserable season is compounded by losing their Scottish Cup stranglehold to their biggest rivals and a summer clear-out awaits.
Dominic McKay, who watched from the Ibrox stands as he prepares to officially start work as Celtic chief executive on Monday, has a busy time ahead.
What did they say?
Rangers manager Steven Gerrard told Premier Sports: "We rode our luck at times but I don't think anyone could take it away from us that we deserved it.
"We won the league really early and from my point of view I've seen the boys come off it, but it was important we made a statement today."
Celtic interim manager John Kennedy told Premier Sports: "We were too soft in our pressing in the first half. Second half, we reacted well and were on the front foot. But the big chances we had, we weren't clinical.
"In front of goal we didn't have enough belief. When it stays 2-0, you have a mountain to climb."
A return to Premiership action beckons for both sides in midweek. Rangers are away to St Johnstone (19:00 BST) in a dress rehearsal for their cup quarter-final, while Celtic visit Aberdeen (19:45).
- 31BarisicBooked at 85mins
- 17AriboBooked at 74mins
- 20MorelosSubstituted forRoofeat 80'minutes
- 14KentBooked at 90mins
- 93LaxaltSubstituted forTaylorat 45'minutes
- 17ChristieSubstituted forGriffithsat 73'minutes
- 8BrownBooked at 47mins
- 27ElyounoussiSubstituted forAjetiat 86'minutes
- Bobby Madden
- Shots on Target
Second Half ends, Rangers 2, Celtic 0.
Attempt missed. Leigh Griffiths (Celtic) left footed shot from the right side of the six yard box is just a bit too high following a corner.
Corner, Celtic. Conceded by Connor Goldson.
Ryan Kent (Rangers) is shown the yellow card.
Albian Ajeti (Celtic) wins a free kick on the left wing.
Foul by Connor Goldson (Rangers).
Substitution, Celtic. Albian Ajeti replaces Mohamed Elyounoussi.
Borna Barisic (Rangers) is shown the yellow card.
Corner, Rangers. Conceded by Stephen Welsh.
Corner, Rangers. Conceded by Greg Taylor.
Offside, Rangers. Joe Aribo tries a through ball, but Kemar Roofe is caught offside.
Attempt saved. Kemar Roofe (Rangers) right footed shot from outside the box is saved in the top left corner. Assisted by Glen Kamara.
Substitution, Rangers. Kemar Roofe replaces Alfredo Morelos.
Foul by Mohamed Elyounoussi (Celtic).
Connor Goldson (Rangers) wins a free kick in the defensive half.
Penalty saved! Odsonne Edouard (Celtic) fails to capitalise on this great opportunity, right footed shot saved in the bottom left corner.
Penalty Celtic. Leigh Griffiths draws a foul in the penalty area.
Penalty conceded by Joe Aribo (Rangers) after a foul in the penalty area.
Attempt blocked. Odsonne Edouard (Celtic) right footed shot from outside the box is blocked.