Danny Ings capped a lively Liverpool start by tapping in from two yards out after being teed up by Adam Lallana.
Rodgers's side were unable to capitalise on their early supremacy and were made to pay before the first half ended by Carlisle.
Liverpool had an early second-half penalty appeal rejected as substitute Divock Origi went down under an Antony Sweeney challenge.
Origi then tested Keith Gillespie in the Carlisle goal before Ings fired into the side-netting from close range.
Rodgers introduced Philippe Coutinho midway through the second half and the Brazil international fashioned a chance for Lallana before firing a 25-yard strike just wide of the post.
It was Ibehre who had the first notable chance of extra time as he skipped between Martin Skrtel and Can before seeing Bogdan save his strike.
James Milner went closest for Liverpool in the first period of extra time as his effort, which was arrowing towards the bottom corner, was blocked by Gary Dicker before Origi's rebound was deflected over the bar.
Rodgers's side saw another chance missed in the second half of extra time as Dejan Lovren directed a header just wide of the post, before Can drove a shot just over the crossbar not long before the end of the extra 30 minutes.
The Croatia centre-half was stretchered off with three minutes remaining in the tie, which saw Keith Curle's men almost win at the death as on-loan Newcastle United midfielder Alex Gilliead saw his 40-yard effort go just wide.
The penalty shootout saw Milner and Can score either side of a Danny Grainger effort which was saved by Bogdan, before Dicker converted for Carlisle.
Ings then scored Liverpool's fifth spot kick, firing straight down the middle past Gillespie, before Hery's effort was denied once more by Bogdan to see Rodgers's team advance to the fourth round where they will take on Bournemouth at Anfield.