The Merseysiders took the lead in the 36th minute when Ryan Shawcross headed Steven Pienaar’s cross into his own net, before Leon Osman spurned a wonderful opportunity to double their advantage when he fired wide.
Stoke then equalised through Kenwyne Jones in the 52nd minute to earn a point for the Potters and move them up to ninth place in the Premier League table.
“We dug in and got a hard-fought point,” Moyes told BBC Sport.
“Stoke are a good team, they make it difficult and anyone who comes here will find it difficult but we had a wonderful chance to make it 2-0.
“Leon Osman missed a great chance, we should have stopped their goal and we were a bit fortunate that Stoke did not get the winner.”
Meanwhile, Stoke boss Tony Pulis was left furious after the officials missed Marouane Fellaini’s headbutt on Ryan Shawcross before an Everton corner.
“I’ll let David [Moyes] deal with Marouane Fellaini but the one thing I will say is we have now had three incidents this season,” said Pulis.
“David Luiz for Chelsea made a poor challenge and Ciaran Clark at Aston Villa made a poor challenge and that, along with this today, are three incidents where we haven’t had a decision go for us and that’s disappointing.
“In my opinion all three of these incidents should been sending offs. I like Mark [Halsey], he is a good referee but maybe he has not seen it.”
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news