Everton started well, with Marouane Fellaini testing Michel Vorm inside 20 seconds and Nikica Jelavic firing wide minutes later when clean through.
Michu came close to netting his 14th Premier League goal of the season when his lofted shot was tipped onto the crossbar by Tim Howard.
Phil Neville and Phil Jagielka both shot narrowly wide in the second half, before Belgium international Fellaini drew a top save from Vorm in the closing moments.
“We missed a lot of chances,” Moyes told BBC Sport.
“On another day, if we’d taken one then we’d have won the game. We could have played better, but Swansea deserve a lot of credit for what they’re doing.
“It’s a great achievement by my players [three defeats in 31 Premier League games] but what happens when you get on a run like this is that you want to win every game. However, I can’t fault the players for their efforts today.”
Meanwhile, Swansea manager Michael Laudrup urged Spain boss Vicente Del Bosque to call-up Michu after the World Cup-winning coach watched the striker at Goodison Park.
“Obviously this wasn’t the best game to see Michu in because it was a very difficult game to show what he is capable of doing,” he added.
“But I know Del Bosque has other people over here who can tell him how he has been playing. I hope he can get to play for Spain, who are the best team in the world right now.”
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