Having twice surrendered the lead, including in the 123rd minute, at Anfield, the Reds eventually prevailed 14-13 in a penalty shootout to overcome their Championship opponents in the third round of the competition.
Liverpool had unexpectedly suffered defeats in two of their previous three games, and Rodgers has praised the winning mentality of his players and the quality of their spot kicks in ensuring that they did not end up on the losing side again.
"It was very important," he told reporters. "You can see how happy the players are. They are competitive players, they want to win in every competition.
"Tonight, a lot of the players haven't played a great deal. Jose Enrique, I thought, was outstanding, he got through the whole game and extra time and showed good composure on his penalty. We're all very happy. We're in the hat for the next round, that's all we want.
"It was a great quality of penalty, I thought both teams showed great composure. We're obviously very pleased to get through after being in the lead and giving away poor goals, to be honest. At the moment we're making too many mistakes, but I can't fault the character and the mentality of the players because they kept going."
Teenage debutant Jordan Rossiter put the Merseysiders ahead on 10 minutes, before Adam Reach equalised. Suso restored their lead in extra time before a Patrick Bamford penalty levelled things up again.