The Dutchman parried spot-kicks from Kevin De Bruyne and substitute Sergio Aguero before a jaw-dropping finger-tip touch flicked De Bruyne’s long-range rocket onto the post.
City did find a way past Stekelenburg through Nolito’s header, but only after Romelu Lukaku had put Everton ahead.
Former Toffees man Stones said: “I think we should have had the three points in the bag but it’s credit to Everton and Stekelenburg - he pulled off some great saves, two penalty saves, which it made it difficult for us.
“As you said I thought we did everything we could, played some great football at times but we got done on a sloppy counter. It’s one of those games and hopefully we can move on from it and take it onto the Champions League on Wednesday.
“I’ve played a lot of games with Rom and his goal tally speaks for itself really, he’s clinical when he gets in the box and he scored a lot of goals so we tried to do our best in stopping that but he got one against us.
“We’ve gone on such a great run, we still got a point, it's the least of what we wanted, I thought we should have had three.
“In the last three games it's two draws and a loss but before that we were on such a great run. For everyone that’s seen the game today, the attacking threat we had, the excitement, I think it’s good for the future and hopefully we can get back to winning ways.”