Henderson opened the scoring with his second goal in as many matches, thrashing home from the edge of the penalty area to give Liverpool a richly deserved 29th-minute lead.
The stand-in captain then turned provider, crossing for Daniel Sturridge to head home just seven minutes after the break.
Henderson pulled the strings throughout and Rodgers was delighted with his performance.
"His strike was terrific and that obviously gives you greater confidence in the game," Rodgers said.
"His ball for the second goal was terrific. He's got great technique and precision.
"I think he's improving all the time. As he matures more, tactically, he'll become even better. He's always had athleticism, he's born with a natural gift as an athlete, to run.
"Tactically, he's improving. Technically, as you see, he is quality. His passing has improved.
"He's becoming one of the real leaders of this young team. Wonderful performance by him."
Rodgers felt the win was just as impressive as Liverpool's 2-1 triumph over champions Manchester City on Sunday.
The Anfield outfit are now unbeaten in 12 Premier League matches and have risen to fifth in the table.
"I thought it was an excellent performance," he added.
"I think Burnley have showed this year that they're a very difficult team to play against, especially when they are on their travels.
"As good as the performance was on the weekend, this was equally as good, in a different way.
"Organisation defensively was very good. Another team that have had no shots on target. And they have got players who can cause you a problem."
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