Leigh Griffiths' late strike was enough to beat St Johnstone and book holders Celtic a place in the semi-final of the Betfred Cup against Hearts.
The Scotland striker had passed up three good chances in a game that came to life only after the break.
However, with seven minutes remaining he pounced from close range to fire in for a 1-0 win and ease recent pressure on the Parkhead men, albeit defender Dedryck Boyata was given a red card in the 90th minute by referee John Beaton for what appeared to be dissent.
Boss Brendan Rodgers, who revealed before the game that there was interest in him from a Chinese club in the summer, had come in for some criticism after Sunday's 2-1 defeat at Kilmarnock confirmed the Hoops' worst start to a Ladbrokes Premiership season in 20 years but Griffiths' goal offered respite and keeps hope of a treble-treble alive.
Rangers clinched their place in the Betfred Cup semi-finals with a comfortable 4-0 win over Ayr United - but now face a Hampden dilemma after frontman Alfredo Morelos picked up a booking that will rule him out.
Gers' top scorer netted the third goal against the Ladbrokes Championship leaders as the Light Blues strolled into the last four.
But another petulant outburst at referee Bobby Madden earned him his second caution of the competition, leaving Steven Gerrard with a major headache ahead of their semi showdown with Aberdeen.
With Kyle Lafferty cup-tied, the Ibrox boss will have to come up with a new solution.
Teenager Glenn Middleton, however, showed he might just be the man to spearhead Gers' attack at the national stadium with a well-taken brace on his full debut in the first-team.
Gerrard made six changes to the side that dismantled St Johnstone on Sunday but they showed no sign of disruption after Nikola Katic nodded them ahead early.
Olly Lee and Steven Naismith scored in the closing stages to send Hearts to Hampden after a pulsating Betfred Cup quarter-final against Motherwell.
Lee's 88th-minute strike put Hearts back in front and Naismith sealed a 4-2 win in stoppage-time when he netted on the break as Motherwell pushed to take the game into extra-time.
Ryan Bowman had threatened to do that with a brilliant finish after close-range finishes from Steven MacLean and Peter Haring had cancelled out Curtis Main's early opener from the spot.
Hearts will face holders Celtic in the last four, while Rangers will take on Aberdeen in the other semi-final.
