Milner returned to the side after missing out against Bayern Munich in midweek, replacing Trent Alexander-Arnold at right-back.
He delivered an accomplished defensive performance, although he failed to alter much of Liverpool's woes on the day as their attacking trio misfired.
Liverpool still go back to the top of the Premier League table, holding a one-point lead over rivals Man City.
However, Milner believes Liverpool could have done more to exploit United's weaknesses, insisting the Reds need to "keep going" and improve whilst in possession.
“I'd say a clean sheet is pleasing, [it is] not an easy place to come anyway,” he explained.
“We did all the running, but didn't create as many chances as we've liked.
“The least we wanted was a clean sheet, [but it is] disappointing not to win. We'll know if it's a good point after 38 games.
“I don't think we were good enough with our movements, the final ball at times, around the box we didn't manage to open them up as much as we'd have like.
“They're a confident team and well organised, but with the players we've got we'd have liked to have made more opportunities.
“After every game, when you play for a team like Liverpool, you want to win and we know we can.
“When you come off it's disappointing, but we've got to hope it's a good point at the end of the season.
“We just have to keep going. We've got to look where we can improve when we've got the ball. We've come up against it a lot this season.”