Robert Huth scored the only goal of the match, turning Dean Whitehead's 15th-minute shot past Tim Howard.
Marouane Fellaini, the Everton midfielder, should probably have had a penalty when he was manhandled by Ryan Shawcross in the area just before half time but Mason, despite being in a good position, did not see it.
Stoke's physical approach to defending, especially in their own 18-yard box, led to frustration for the Everton players, but Moyes acknowledged they had not done enough to force an equaliser.
"I thought the referee had a poor game," said the Scot. "It is not the reason for our loss - far from it because that was down to our inability to make and take chances.
"But there was a lot going on in the box and we got very little of it."
In a match of few chances Everton enjoyed more possession but failed to trouble Thomas Sorensen and, when he was carried off on a stretcher after a clash with Tim Cahill late in the second half, his replacement Asmir Begovic had little to do, too.
"We huffed and puffed a little bit but I don't think we conceded many chances to Stoke," Moyes said. "We weren't peppered and lost the game that way. We defended a corner but didn't defend the edge of the box and they get the shot and the flick in.
"We hadn't done an awful lot wrong and we found ourselves a goal down. I don't think we deserved to be behind in the game but we had to go chasing it.
"What wins games is goals and they got the goal and we didn't."
Moyes gave the 19-year-old striker Apostolos Vellios only his second start for the club after Louis Saha pulled out with injury.
Stoke made it back-to-back league wins, sandwiched between a Europa League draw at home to Dynamo Kiev.
"This was a real battling performance against a good Everton team," said Tony Pulis, the Stoke manager. "It has been a great week for the club. To get six points and to qualify for the latter stages of the Europa League is first class.
"We have had a dip of form, which most clubs do, and we have shown a lot of character and spirit the last week."
Pulis said Sorensen suffered no lasting effects from his clash with Cahill. "Thomas has mild concussion. The doctor's looked at him and he's OK," he added.
* Press Association
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