Rodgers saw captain Steven Gerrard score twice from the penalty spot at Old Trafford before Nemanja Vidic was shown a second yellow card late on for a foul inside the area on Daniel Sturridge.
Gerrard failed to convert his third penalty, but there was still time for Luis Suarez to find the target as Liverpool completed a fine win to close within four points of Premier League leaders Chelsea, who have played a game more.
Rodgers felt Liverpool could have scored more goals but was keen to reiterate that nobody at the club is getting carried away with their title challenge.
"It's obviously a big win over a huge rival - two of the biggest British clubs that are renowned worldwide. To dominate the game like that was very pleasing," he said.
"I thought we could have had more goals, I thought we were really aggressive in our game going forward and with a better final touch or final ball we might have got in a bit better and finished them off even more so.
"We attack with great intent and we defend as well.
"We will just focus on our next game. I can only look at our next game. I've done that since I walked in the door."
As he approaches the end of his second season at Liverpool, Rodgers added that he expects his charges to improve as the campaign draws to a close.
This, he explained, is largely due to a change in mentality at Anfield as Liverpool eye their first league title since 1990.
"We were the same last season - we get better with the methods and that gives us confidence and belief. I try to not expect so much and let the players do the talking," he added.
"We're up there on merit so of course we have an opportunity.
"We expect to win. We have a mentality in the team that's been developing over the course of the 18 months I've been here that we expect to win, home and away.
"The belief is in the players now and you see that in their game. I haven't banned (talk of winning the title), we've just got better things to talk about.
"I don't lose sleep over it, (we) just talk about players improving their performance level and improving their life."
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