The in-form Reds had only lost once in 15 games coming into the match but looked at risk of suffering another defeat after finding themselves trailing at half-time at Selhurst Park to a Fraizer Campbell goal.
However, fit-again striker Daniel Sturridge drew the teams level soon after the break and then Adam Lallana scored what would prove to be the winner after following up a Mario Balotelli free-kick to net on the rebound.
Rodgers told BT Sport: “We had to be patient but of course we had to show character, and the character in the team has been immense in the last few months.
“Once we got the (equaliser), we got our confidence back into the game again.”
Rodgers was also pleased to have “natural” goalscorer Sturridge available again after injury, and believes “the penny has dropped” at last for Balotelli.
The Italy international has largely failed to impress since moving to Anfield last summer but he scored the winner against Spurs last weekend and had a hand in Liverpool’s decider against Palace.
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Rodgers said: “He’s been working really hard in training and I think now the penny’s dropped… I assess players in training every day, no matter how good they are, and they have to put the work in.
“It has been difficult for him, because he’s come into a style maybe he’s never been in before, but one thing he has got is quality, and he’s got big quality and the last couple of games he’s come off the bench and been effective for us.”
Palace manager Alan Pardew described the cup exit as an “opportunity missed” but felt Liverpool deserved their win.
He said: “I felt we didn’t put enough pressure on Liverpool today. We had a great chance at half-time. We should have done better today.
“I’m not too disappointed but I do feel it was an opportunity missed.
“But we’re not going to make excuses and I think Liverpool deserved to win, and you have to accept that sometimes.”
Fancy getting your hands on a signed Steven Gerrard Liverpool shirt? We thought so too. The guys from INPLAY have just the opportunity for you…
INPLAY is a fantastic new social messaging app with a unique football twist, allowing fans to interact with their friends and supporters-alike with customisable chat tools like stickers and the option to share images and videos.
There’s also personalised trending football stories filtered from over 2000 sources that can easily be shared with your football friends.
Here at This Is Anfield, we’ve been using the app ourselves recently and have been very impressed at INPLAY’s growing community, broken into three sections:
CHAT — a multimedia rich messaging area for private chats, club specific or football-wide debate and discussion (lots of fun!)
HOTWIRE — a brilliant, comprehensive news aggregator bringing you the latest football news from across the internet and social media.
DISCOVER — a lively hub for users to find chats, Fan Clubs and like-minded fans.
Now for the competition! INPLAY want to give one passionate Liverpool fan a signed Gerrard home shirt.
The competition runs from until 6 March 2015 and during that time, fans simply need to download and sign up to the INPLAY app (it’s free!), join the This Is Anfield Fan Club on the ‘Discover’ tab and get involved with the Liverpool chat in there.
The fan who makes the most interesting and positive contributions over the next few weeks will take home the signed Gerrard shirt, so get involved, invite your mates and get chatting. It’s as simple as that.
Get started now and don’t forget to launch the app for in-match discussion during Liverpool games!