Liverpool headed into the clash at Turf Moor without a win in three Premier League outings, and the home side dominated the opening half, striking the post through Danny Ings.
Rodgers' men survived until half-time, and found the winner shortly after the hour mark when Raheem Sterling rounded goalkeeper Tom Heaton and slotted home.
And the Northern Irishman hailed Liverpool's battling display as they moved up to ninth in the table.
"I thought in terms of our character and resilience in the game we showed outstanding qualities when needed," he said.
"They pressed really hard and made it difficult in the first half but I felt we were better second half.
"I was really pleased because we had to show a different side to us. It's a great three points and I'm delighted with that."
Rodgers also reserved praise for Simon Mignolet who, after being dropped "indefinitely" two weeks ago, came on as an early replacement for the injured Brad Jones, and successfully kept the home side at bay.
"He's done well. He came for a couple of crosses really well," Rodgers said.
"He's worked hard since he's come out of the team, he's kept the clean sheet and that was important for us."
Match-winner Sterling was also lavished with praise, after the England forward had come in for criticism following a wasteful performance in the 3-0 defeat to Manchester United earlier this month.
"He's in a position now where he's going to receive opportunities," Rodgers said.
"He's going to get those central runs and since that game he's come away and worked on that type of finish.
"He showed great composure and great ability to learn. He finished it really well so his development has got us the three points."
