Rodgers, taking charge of his 50th Premier League game at Liverpool, saw his side twice take the lead before a Romelu Lukaku double put Everton in front with eight minutes remaining.
However, substitute Daniel Sturridge secured a point for the visitors with time running out and Rodgers was happy with the result as well as the determination his side displayed.
"When you come to your rivals, it's very important that you don't lose," he told BT Sport. "Everton have a great home record, you have to show resilience and character, and we showed that.
"We can score goals – we showed that again today.
"Overall I'm delighted with the character of the players and (it was) a good point away from home. There's a hell of a long way to go and there's some great teams up there, but we feel like we're improving all the time."
Rodgers also felt Kevin Mirallas should have been dismissed for a challenge on Luis Suarez in the first half.
The Belgian was shown a yellow card for a knee-high tackle on the Liverpool striker, with Rodgers claiming the decision may have been different if Suarez was the aggravator.
"I thought Mirallas should have went," he said.
"It would have been interesting if Luis Suarez had made the challenge. It was a poor, poor challenge. Those are career-threatening challenges. It should have been a red."
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