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, with Rodgers praising his side's defensive display and hailing the performance as evidence of their contrinued development.
|12/1||Liverpool are 12/1 with bet365 to win the Premier League
"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 match-winner Sturridge, who grabbed the only goal of the match despite being "nowhere near" full match fitness.
"[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."