The Merseysiders went to the top of the Barclays Premier League with a third consecutive one-goal victory while United now have a mix of a win, draw and loss from the three matches ahead of the international break.
"I thought we played very well from the start," he insisted. "We gave away a bad goal in the opening few minutes but, with the exception of that, it was probably the best we have played this season.
"We just didn’t quite have the last bit which would have enabled us to score. We dominated the game for long periods without us having the final ball or cross and, when we did get them, it just didn’t quite fall for us.
"It wasn’t for the want of trying. It wasn’t for the amount of forward players on the pitch to try and get them to score that’s for sure. What I would say is we didn't have Wayne Rooney because of injury and I'm aware we lacked a bit of something different at times.
"Liverpool had very few opportunities. They had two free-kicks outside the box, Steven Gerrard had a shot and there was a chance in injury time. We gave the ball away for the half-chance which led to the corner that they scored from, so that was disappointing. The players know that and aware of it."
The tough start was always going to be a major test for the Reds and the small matter of the Manchester derby remains on the horizon.
"I always knew it was going to be a tough ask," insisted Moyes. "No matter if Sir Alex was sitting here playing these games. It's undoubtedly a