Daniel Sturridge's fourth-minute strike was all that separated the two sides in a hotly contested local rivalry to maintain Liverpool's winning start to the season.
United never really threatened the Liverpool goal and Simon Mignolet was only forced into two saves as he kept his third successive clean sheet.
The 40-year-old praised his side and felt their performance was evidence of their continued development.
"I thought the result was outstanding, and defensively I thought we were very, very good," he said.
"We'll be more fluent and as the season goes on we'll become better, but these are the big games we wanted to be winning this year.
"I think you see the togetherness and spirit this year, to get three wins out of three I can only be happy with."
Rodgers was particularly impressed with his match-winner Sturridge, and his overall performance.
Sturridge - who turned 24 on Sunday - reacted quickest to divert a Daniel Agger header into the back of the net and provided a constant threat to the United defence throughout the match.
"(I'm) delighted with his application because he was nowhere near fit today," he added. "He played a full game and extra time in the week and had slight knock on his thigh.
"Even at 70 per cent he'll be as good if not better than a lot of Premier League strikers.
"(He showed) great anticipation coming off the keeper, and when he's on the pitch he can be a threat so there's no doubting his qualities."
Liverpool travel to Swansea in their next Premier League match after the international break.
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