The north west rivals drew 1-1 at Anfield on Saturday after Javier Hernandez's late header cancelled out Steven Gerrard's second half free-kick.
Javier Hernandez's 81st minute header secured a draw for United at Anfield after Liverpool captain Steven Gerrard had given the home side the lead with a free-kick in the 68th minute.
Liverpool created the better chances and applied late pressure as the team tested United goalkeeper David de Gea, but it could not score a winner past United's stubborn defense.
Dalglish took charge of Liverpool in January and he believes it is a sign of the improvement his side has made in recent months that the players were downbeat after being unable to seal victory over the Premier League champion.
“We could have had all three points after what I think it was pretty even and sterile beginning to the match,"Dalglish said after the match. "I think it's a real indication that the players were disappointed in the dressing room after the match about drawing with a team like Manchester United. That shows how far we have come."
The Liverpool boss went on to praise the play of De Gea, who has had his struggled early in his United career.
"I saw in the press that De Gea has struggled, but he certainly didn't struggle today, he made a couple of fantastic saves," he said.
Gerrard's goal marked his return to the Liverpool starting lineup after a troublesome groin injury and Dalglish was delighted to see his skipper get his name on the scoresheet.
“It's fantastic for Steven that he has crowned his first start for a while by scoring,” Dalglish said. “It's brilliant to see him back. Everyone is happy with how far we've come and wants to take it further. I'm sure Steven can help with that."
United manager Sir Alex Ferguson surprised many by naming Hernandez, Wayne Rooney, Nani and Anderson as substitutes in a defensively minded starting group. Dalglish, however, interpreted his compatriot's team selection as a compliment to Liverpool and acknowledged rotation as a large part of modern management.
“Their team selection was not a surprise,” he added. “I think it was a compliment to us as well. Fergie always changes him team after an international break and they have got an important game on Tuesday. I've always said it's a squad effort and obviously Sir Alex thinks the same way.”