The Toffees manager was frustrated that his side were unable to take any points away from the clash after they missed a number of opportunities in the first half at Goodison Park
The Toffees dominated the match for the first 25 minutes and took the lead through Steven Pienaar but a Frank Lampard double either side of half-time sunk the Merseyside club.
Moyes claimed he was not disappointed with his team's performance but rued the chances they missed and suggested that fortune favoured the visitors.
"We're not disappointed with the way we've played because we played really well and we've run the European champions really close," Moyes told Sky Sports. "But we didn't quite have enough in the end.
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The Scot has had several issues with injuries to first team players, and admitted that the absence of the likes of Phil Neville, Darron Gibson and the suspended Marouane Fellaini hurt his side's chances.
"We tried to make what we had work the best we could today and for long periods I thought we did, we played well enough to get something from the game," he added.
"We were limited today with what we had, with Phil Neville and Darron Gibson out injured and Marouane Fellaini suspended. But Nikica [Jelavic] had a great chance to make it 2-2 in what felt like the 94th minute."The Everton boss was unable to hide his frustrations after the final whistle, when he confronted referee Howard Webb regarding his failure to award Leon Osman a free kick after it appeared the midfielder was tripped by Lampard.
"I thought it was a free kick on Leon Osman on the edge of the box in the 92nd minute by Lampard and he didn't give it," Moyes said. "But I've apologised to Howard Webb because I shouldn't do that coming off the pitch."