After Danny Ings’ second goal in as many games for the Reds was cancelled out by Derek Asamoah in the first half to see the scores level at 1-1 after 90 minutes, the match eventually went to spot-kicks where debutant Adam Bogdan was their saviour with three saves.
But the cheer around Anfield as the Hungarian goalkeeper denied Bastien Hery was more relief than anything else.
Liverpool earlier in the day had to deny reports they had approached Carlo Ancelotti about taking over at Anfield, with Rodgers still under pressure.
He is without a victory in 90 minutes in the last six games and although he named a side which contained five of the team which drew against Norwich at the weekend — plus goalscorer that day Ings — they struggled to breach the leakiest defence in League Two, despite having 47 shots in 120 minutes.
Liverpool have not kept a clean sheet at home in the League Cup since January 2005 against Watford and that was to prove costly, as was a record which shows only twice in the last 21 matches have they scored more than once in a game — the last time coming on May 2 against relegated QPR.
Ings, as at the weekend when he came off the bench to score, was a bundle of energy and his tenacity and running always looked like creating something.
Having seen Mark Gillespie parry one effort — James Milner heading wide the rebound — and looping a second wide, he got his rewards with a close-range header from a well-timed run to meet Alberto Moreno’s cross.
The same combination almost produced a second but the former Burnley striker nodded wide — although not as wide as the gap Liverpool afforded Hery as he skipped unchallenged through central midfield to slide the ball through to Asamoah, who beat Bogdan at his near post.
Forward Roberto Firmino, struggling with a back injury after falling awkwardly, was immediately replaced by Divock Origi and that affected Liverpool’s momentum almost as much as the equaliser, as Ings, the firebrand up front, dropped back to a deeper role and lessened his threat.
Anthony Sweeney’s second-half challenge on Origi was probably worthy of a penalty, with the Belgium striker and substitute Philippe Coutinho also forcing good saves out of Gillespie and Milner’s strike deflecting into the side-netting.
Carlisle somewhat generously waited until the 64th minute before introducing Jabo Ibehre, with 10 goals in 11 games this season, but with Liverpool laying siege to their penalty area they were restricted to just a couple of opportunities.
Dejan Lovren’s departure on a stretcher with his right leg in a splint added to Liverpool’s woes as they finished the game with 10 men, having used all their substitutes.
Penalties ended the agony, with Milner, Emre Can, and Ings all scoring against the two successes from Gary Dicker and Alex McQueen — but victory posed more questions than answers and there is more scrutiny to come for Rodgers.
Bogdan, Can, Skrtel, Lovren, Clyne (Ibe 86), Milner, Allen (Coutinho 64), Moreno, Firmino (Origi 35), Ings, Lallana.
Gillespie, Miller, Raynes, Grainger, McQueen, Dicker, Joyce, Kennedy (Gillesphey 73), Sweeney (Ibehre 64), Hery, Asamoah (Gilliead 64).