Having taken an early lead at the Etihad Stadium on Tuesday through a Sergio Aguero penalty, the Premier League champions were pegged back as Francesco Totti produced a stunning chip over Joe Hart just before half-time.
City's efforts to get back into the game saw them create plenty of opportunities after the break, but only tested Roma's back-up keeper Lukasz Skorupski on two occasions.
Milner, a second-half substitute, admitted City's normal pace in attack had been missing in their Group E clash, and rued a missed opportunity to claim all three points.
"Maybe we could have won it, I don't think we played our best - we know that," he told ITV. "We didn't really have that much tempo in the game but I thought we created a few chances and probably could have nicked it.
"We never got up to our speed. I think that's one of our strengths, when we get the ball moving quickly there's movement everywhere. Teams find it hard to deal with it.
"Well-organised teams come here. We're used to that and we have to move the ball quickly to get past them. They're a well-organised team, a good team and they're used to doing that.
"So we didn't quite to enough to win the game."
City sit third in Group E after the draw with one point from their opening two matches.
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