Steven Whittaker was sent off by referee Calum Murray in the 35th minute after picking up his second booking for a foul on Emilio Izaguirre and consequently an uphill task for the injury-ravaged visitors became almost impossible.
Mark Wilson drove in three minutes after the break after his first shot was cleared off the line by the head of Ibrox defender Sasa Papac – who was stretchered off for his troubles to add to the Light Blues’ woes.
Much of the drama, however, came at the end.
Rangers defender Madjid Bougherra was sent off in injury time after picking up his second yellow and after the game, as the two benches clashed, the unruly El-Hadji Diouf, booked earlier, got another yellow card for approaching Murray and was dismissed.
With victory Celtic booked a place in the quarter-final away to Inverness and will be firm favourites to win the tournament.
Rangers striker Nikica Jelavic dropped out with a knee injury, so Walter Smith decided to go with the same side which battled through against Sporting Lisbon in their Europa League tie last week – and it was a 5-4-1 formation built to defend. David Weir returned to the heart of the Light Blues defence as expected.
Vladimir Weiss, Kyle Lafferty and David Healy dropped out with Richard Foster, John Fleck and Diouf coming in, the latter starting as a lone striker.
Celtic’s changes were more predictable. Lukasz Zaluska and Beram Kayal replaced suspended Fraser Forster and Joe Ledley respectively while manager Neil Lennon drafted in Georgios Samaras at the expense of Anthony Stokes, who dropped to the bench.
In the 12th minute Rangers had half a chance when Diouf raced through wide on the right with Zaluska coming out to leave his goal unprotected. However, although the former Liverpool striker got a cross in from the byline, the danger was easily snuffed out by the backtracking Hoops defence.
For all their possession the home side could not fashion a clear chance.
Skipper Scott Brown, who continued his ongoing feud with Diouf throughout the first half, took advantage of a break of the ball to run through the Ibrox defence but Kyle Bartley had the pace to get back and Allan McGregor smothered.
In the 28th minute Ki Sung-yueng was fouled by Whittaker, which resulted in a yellow card for the Rangers midfielder who was dismissed seven minutes later after picking up another caution card for a reckless challenge on Izaguirre.
Diouf and Hoops boss Lennon then had a touchline confrontation which earned the Light Blues player a booking and which threatened to plunge the game in to further disorder.
Referee Murray then spoke to Rangers assistant manager Ally McCoist and Lennon – a spat which would resume later.
When play resumed a mistake by Ibrox midfielder Maurice Edu helped Samaras go through against McGregor – but the Rangers goalkeeper made a great block from the striker’s close-range shot.
There was another fracas between Hoops’ assistant Johan Mjallby and Bartley as the players made their way up the tunnel at the interval, with Diouf involved again.
When Wilson scored three minutes after the break it put the home side firmly in the driving seat.
Izaguirre’s deep cross from the left found his fellow full-back at the far side of the box and his first shot was headed off the line by Sasa Papac – but the ball rebounded back to the former Dundee United player whose second effort found the target.
Papac failed to recover and was carried off on a stretcher, replaced by Gregg Wylde.
The game became increasingly antagonistic and Bougherra was cautioned for a foul on Gary Hooper which left the Celtic striker requiring treatment.
The Parkhead side were in complete control and in the 63rd minute Samaras crashed a low drive off McGregor’s left-hand post.
Izaguirre then set up the former Manchester City with another fine cross but his weak header was easily saved by McGregor.
Deep into injury time Bougherra was dismissed after fouling commons – and there was more mayhem to come.
After the final whistle Lennon and McCoist had to be separated before Diouf received a second yellow and a subsequent red after approaching referee Murray.