Celtic eased holders Rangers out of the CIS Insurance Cup to heap more pressure on beleaguered Ibrox boss Alex McLeish.
The home side were already in control when Shaun Maloney sidestepped his marker and drove the ball from 30 yards past a badly-positioned Stefan Klos.
Rangers defender Sotirios Kyrgiakos was sent off after sarcastically clapping the officials following a foul.
And Klos diverted a Shunsuke Nakamura cross into his own net under pressure from Bobo Balde to end Rangers' hopes.
It was sweet revenge for Celtic after their 3-1 defeat by their city rivals in the league in August.
Celtic boss Gordon Strachan was under pressure at that time, but it was McLeish who went into the quarter-final with his job on the line after just two wins in Rangers' last 10 matches.
The Rangers manager was also forced to introduce Hamed Namouchi and Ian Murray into midfield with Peter Lovenkrands only fit enough to be on the bench after a virus and Brahim Hemdani injured.
Unchanged Celtic not surprisingly started the more confident side.
And Aiden McGeadie, who continued in midfield despite the return from suspension of Alan Thompson, tested Klos with a drive and just failed to find John Hartson with a cross to the back post.
Celtic broke the deadlock with a superb strike by Maloney from 30 yards that Klos, who looked slightly out of position, got a hand to but could not prevent bulging the net.
Rangers were failing to penetrate the Celtic defence and resorted to long-range efforts.
Namouchi's low drive flew a couple of yards wide, Barry Ferguson fired straight at Artur Boruc and Fernando Ricksen curled an effort a few feet over the bar.
McLeish replaced Oliver Bernard with Lovenkrands at the break in a more offensive formation.
And McGeady almost presented Rangers with an equaliser with a slack backpass that Balde failed to control.
But Dado Prso sliced the ball wide from 16 yards under pressure from the big defender.
Rangers chances virtually ended when Kyrgiakos was shown a second yellow card for dissent after he was ruled to have fouled Maloney.
Stilian Petrov had a goal chopped off for offside, Hartson missed badly from six yards and Nakamura forced a fine save from Klos.
But the German goalkeeper completed a nightmare night with an own goal before Rangers came close to a consolation when Prso headed on to the crossbar.
Celtic: Boruc, Telfer, Balde, McManus, Camara, Nakamura, Lennon, Petrov, McGeady (Varga 77), Hartson, Maloney.
Subs Not Used: Marshall, Thompson, Virgo, Pearson.
Booked: Hartson, Balde.
Goals: Maloney 26, Klos 82 og.
Rangers: Klos, Hutton, Rodriguez (Pierre-Fanfan 79), Kyrgiakos, Bernard (Lovenkrands 45), Namouchi, Ferguson, Ricksen, Murray, Prso, Thompson (Jeffers 56).
Subs Not Used: Alex Rae, Waterreus.
Sent Off: Kyrgiakos (66).
Booked: Kyrgiakos, Hutton, Jeffers, Pierre-Fanfan.
Att: 57,183
Ref: K Clark