A deflected Charlie Adam free-kick put the home side in front on 38 minutes, with the Reds also losing both Jonny Evans and Phil Jones to injury by half-time.
Robin van Persie's goal soon after the restart restored parity, but an unstoppable effort from Adam and some goalkeeping heroics from Asmir Begovic right at the death secured the victory for Stoke.
"We played well today," Moyes told MUTV. "We’re definitely not getting a lot of good fortune going our way, but we’ve got ourselves to blame for the opportunities we missed. I thought we played some good football, we got to the byline, but just couldn’t pick anyone out in the box.
"I think it [the deflected free-kick] summed it up. That went in after we’d controlled the majority of the game. I think it was going wide, but it took a wicked deflection and went in. That’s when you know that your luck isn’t in. It’s just not going for us.
"It was a difficult day, the conditions were tough," added the manager. "We had opportunities to get at least a draw and they scored a world-class second goal which was hard to defend against."