Robin van Persie gave United a first-half lead and Nemanja Vidic extended their dominance in fortuitous fashion from a free-kick early in the second half.
But substitute Daniel Sturridge's first league goal for Liverpool brought the visitors back into the game and there were one or two nervy moments for the hosts in the closing stages.
Rodgers said: "I was very proud of the team. I thought that the first half we were a wee bit tentative at times in our passing and we actually presented them with opportunities in the game
"But in the second half - you could have easily at 2-0 given up the fight - the fightback was brilliant by the players and I thought second half we maybe deserved something from the game.
"We are disappointed to lose but take heart from the fact we kept going and Daniel Sturridge coming on and getting another goal was terrific. The second-half performance was much better.
"It's never easy when you have got young players that are making their debuts at Old Trafford and when you look at Manchester United's experience in their team - they have been there and been around a lot of them.
"For us it was just about reinforcing a couple of details that we needed to be better at. We are disappointed with the free-kick because we never defended that well enough. But after that the reaction was terrific.
"We will take great hope from the second-half performance. We know the improvements we need to make if we are going to sustain a challenge at the top of the table.
"There was 21 points difference before this game but it wasn't 21 points in terms of quality, it was 21 points in terms of the depth of the squad. So if we are going to be able to compete we need depth in our club that is going to enable us to compete on a consistent basis."
Rodgers also paid tribute to Sturridge who entered the action at the start of the second half.
He said: "I thought he was a real threat when he came on. He is an outstanding young player who just needed an opportunity. He will score many goals for Liverpool I am sure."