A sixth game without a win would only have increased the pressure on the under-fire Rodgers, on the back of a rising tide of resentment from the Anfield faithful.
Bogdan, who was making his competitive debut, stopped three of Carlisle's five penalties after Derek Asamoah cancelled out Danny Ings' opener and the two sides stayed deadlocked after extra time.
"There's method in the madness," the Hungarian told the club website (www.liverpoolfc.com).
"Sometimes you guess right, sometimes you get lucky and sometimes you don't. I guessed the right way and got lucky with three shots. Thankfully we went through."
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Up next for the Merseysiders is Aston Villa's visit to Anfield in the Premier League on Saturday and Bogdan is conscious of how important a victory is.
"All the games are massive," the keeper said. "Every week in the Premier League there is no easy game."
"We have to concentrate on the next one, come in on Saturday, get the job done and win the game and move on."
Recent history seems to favour Villa, as none of the last eight meetings between the teams has ended in a home win.
Rodgers could be without the services of midfielder Roberto Firmino and defender Dejan Lovren, who were both withdrawn with injuries against Carlisle.
Striker Christian Benteke has already been ruled out with a hamstring problem he picked up in the 1-1 draw with Norwich City at the weekend.
(Reporting by Simon Jennings in Bengaluru; editing by Amlan Chakraborty)
