The Reds manager admitted that his side was too tentative in the first half, but believes that the club's display after the break merited more than a 2-1 loss to its rival.
The Reds went 2-0 down thanks to an early goal from Robin van Persie before Patrice Evra's header was deflected in off Nemanja Vidic. Daniel Sturridge got Liverpool back in the game with his first Premier League goal for the club, but the Reds could not find a leveler despite late pressure.
"I thought at the end we deserved something," Rodgers told reporters. "I am very proud of the fight in the team in the second half. We went 2-0 down and when you come here, it's always difficult. Other teams might have crumbled.
"We were disappointed with the first half. I thought we were too tentative. There wasn't much in the first half, but any chances they had were through us giving the ball away. We then go 2-0 behind with again a disappointing goal but from that moment, I thought our efficiency in the game was much better and by the end of the 90 minutes, I thought we probably deserved something. It was unfortunate, but I am proud of the second-half performance."
Rodgers reiterated his belief in Sturridge's potential after the forward followed his debut goal against Mansfield in the FA Cup by pouncing on a rebound to get off the mark in the top flight.
"This is a boy that can be a real top striker in this country," Rodgers added. "He is going to score many goals for Liverpool. You've seen in half a game today and 55 minutes last week [against Mansfield] that this boy is a goalscorer and all he needed was an opportunity.
"He had one goal and was unlucky a couple of times when he tried to organize his feet inside the box and just couldn't quite get a shot off. He is going to be a terrific signing for us."