The Toffees sealed a Merseyside derby FA Cup semi-final date after a hard-fought 2-0 victory away to Sunderland on Tuesday.
And the Scot was delighted with the way in which his squad maintained their level of performance despite their recent exertions.
He told reporters: "I thought there was parts of [the performance] which were very good and parts of it where it looked like there was a bit of mental tiredness and a bit of physical, which I was really worried about today.
"We put near enough the same boys out again, made a couple of changes, and I thought in the main we did the job, albeit I saw signs of fatigue mentally more than physically really.
"We still needed the second goal and at 1-0 I was deciding ‘do I shore it up?’ because I was just thinking that maybe we didn’t have an awful lot left.
"When we got the second goal it made a big difference it sort of put the game to bed for us.
"There were bits of play in the first half where maybe we could have gone on and scored but we weren’t making the final pass with a bit of tiredness in the general play.
"So, a bit mixed but if you’d have [offered] me this result before the game I’d have said great because West Brom have got one of the best records away from home in the Premier League, so I was really wary of that."
The win moved Everton up to seventh in the Premier League, two points ahead of rivals Liverpool, who have a game in hand.
And the 48-year-old admitted his side's performance this season has surprised him after a summer in which the club sold Mikel Arteta and Yakubu without replacing them.
He added:"If you’d said to me at the start of the season after the summer we’d had that we’d be in seventh with seven games to go I don’t think I’d have believed you.
"At the moment we’re in a stronger position than I thought we’d be in but I think the difference has been the boys in January.
"Landon Donovan, Darron Gibson, Niki [Nikica Jelavic], Steven Pienaar and I think that’s why if you look at the team the difference from now till then was noticeable.
"Even to get Steven Pienaar, Niki Jelavic, Darron Gibson on the pitch today has made a big difference to us.
"I’m just delighted that if you’d said to me I’d have been as close to Liverpool I’m not sure I’d have believed you.
"The players have put in a brilliant effort, they’ve tended to come good in the second half of the season. I’m just hoping that’s the way it keeps going."
The Toffees have earned a reputation for starting poorly in the Premier League before finding form during the run-in, a trick they look set to repeat this season.
And the former Preston boss admitted he has no idea why his players tend to perform poorly in the early months of the season.
"If I could work it out it’d be great because I’d probably try to get a good start as well if I knew why the second half happened," he continued.
"I’m just saying we’ve never started well but if it’s finishing well then it leaves everyone with a good taste in their mouth.
Moyes then went on to insist that his side will not be distracted by the upcoming cup clash with Liverpool and hopes to guarantee a top-10 finish.
"We’ve only got one cup game to play so I’ve got league games to play and if we can finish high up then it’d be great," he said.
"I said anywhere in the top 10 for Everton would be an outstanding position with the way the year has been and I still say the same."
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