Saturday's reversal at the Britannia Stadium was United's eighth Premier League loss this season as a Charlie Adam double either side of Robin van Persie's brief equaliser handed Stoke the points.
After Stoke's first league win over United since 1984, Moyes bemoaned the fact his players failed to make the most of the possession they enjoyed in and around the penalty area.
"I thought the performance was really good. I thought we played well," Moyes said.
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"I thought we were the better team. They get a goal, we get back in it, I thought we were the team more likely.
"We only have ourselves to blame. The amount of opportunities we had today we should’ve won the game out of sight. It was a good performance, we just couldn’t take those chances.
"I thought on the day we certainly did enough to get something out of the game."
United saw Jonny Evans and Phil Jones taken off injured in the first half. Evans has a calf strain and Jones suffered concussion after a tangle with Jonathan Walters.
A scan has since confirmed that Jones is "fine", according to Moyes, who felt United were unfortunate to be on the receiving end of a third successive away defeat in all competitions.
"We had bad luck, we really did," he continued before lamenting some of United's decision-making.
"It was our own downfall that we didn't score our chances. We must have got to the byline eight, nine, 10 times but we never pulled the ball back or never picked someone out in the box so it's our own undoing."
Van Persie, Juan Mata and Wayne Rooney all started together for the first time and, despite failing to win, the United manager with pleased with their performances.
"I thought they all played well today," Moyes added. "Robin played well and got us a goal, Juan Mata played well, Wayne played well. We just couldn’t get the second or third goals which were required to get something out of the game."