Liverpool were 2-0 ahead at halftime before the Blues fought back through Idrissa Gueye, though eventually lost 2-1.
Afterwards, Moyes snapped: "A lot of things that weren't so good in the first half, a lot of good things in the second half. We probably just let it get a bit far away from us. The players never gave up. We did some good things but not enough to get the result today.
"There wasn't enough (chances). Liverpool didn't have many more than us but they we e more clinical. They looked more decisive in the final third. Not an easy game for anyone coming here, never mind us.
"We have been here so many times and I don't want to sound like an old fart saying the same things but I found three minutes amazing at the end of the second half. We wanted to play quickly, I have seen lots of teams score from quick free-kick but we couldn't do anything quickly because we had to wait on the referees whistle all the time. I found some of the decisions really strange. We have come to get used to that and everyone is aware of that."
On strong start to season, he said: "We showed signs of what we can do. This might be the hardest away games so we have to accept it but a lot of good things going on. We have won a lot of games away from home and I am hoping we can continue that."
On whether the tackle on Iliman Ndiaye in lead up to first Liverpool goal was a foul, Moyes said: "I have not seen it again I couldn't be sure. I probably would have said it wasn't a foul but there is a lot of other things that I could have said, but I won't if you don't mind.
"The boys never gave up and showed a great deal of resilience."
On Kiernan Dewsbury-Hall's booking for quick free-kick, he added: "The bigger thing is the game seemed to be controlled by the referee, we couldn't start anything quickly. We couldn't play. I have had played plenty of games where teams have taken quick free-kicks and it hasn't been the referees' whistle.
"He never allowed us to get the game under way fast when we wanted to. I found it strange there was only three minutes (of stoppage time). If Liverpool were 2-0 down I'm not sure if three minutes would have been put up."