Juan Mata's double secured Louis van Gaal's side a 2-1 win at Anfield which moved the Red Devils five points clear of Liverpool in the race for Champions League qualification.
The defeat ended the Reds' 13-game unbeaten run in the league stretching back to their loss against United at Old Trafford in mid-December.
Next up for Liverpool is a tough trip to third-placed Arsenal after the international break.
But Rodgers insists his side can still finish in the top four and has challenged his side to respond in the same way they did after the Old Trafford setback.
"We had a 10-game objective that we set. Built into that was that we were not going to win every single game," Rodgers said.
"It's still very much [possible] for us to go where we want to go, which is the top four. We just need to fight that bit harder and make up a five-point difference.
"That last performance at Old Trafford was a great springboard for us. We went on and, up until this point [against United], we hadn't lost a league game.
"So now we must use this similarly. We must use this in the same way; refocus, get over the disappointment and then come back into a tough game next against Arsenal and really look to get the points there."
