While the Reds were being beaten 1-0 at home by Fulham yesterday evening, the Toffees earned a 1-1 draw at Stoke and Moyes' side have gone three points clear of their Merseyside neighbours with two games left each.
It meant Everton, seventh in the table, stretched their unbeaten run in the league to seven matches and, perhaps even more satisfyingly, extended their advantage over eighth-placed Liverpool by another point.
Toffees fans will be eager for their team to better their local rivals' final league position for what would be the first time since 2005, particularly after the Reds' 2-1 triumph in last month's derby FA Cup semi-final.
Moyes, though, has been keen to stress his belief that any kind of top-10 finish would represent a job well done for his men this term, a stance that has not changed.
Asked after the Stoke game about the prospect of finishing above Liverpool, Moyes said: "I feel that if we can just try to keep winning, I will be pleased that we finish in the top 10.
"I think this point might just about get us that. If you said that to me at the start of the season, I would have taken that all day long.
"Certainly our form from January (Everton have lost only twice in 16 league games since mid-January) might be better than just top-10, but at the start of the season, I would have said 'no, I'm happy with that'."
Everton went ahead in the 44th minute at the Britannia Stadium through former Liverpool striker Peter Crouch's own goal and at that stage Stoke had only really threatened with efforts stemming from the long throws of Rory Delap.
But the hosts were reinvigorated in the 67th minute when manager Tony Pulis chose to make a bold triple substitution.
Crouch, Jon Walters and Delap were replaced by Cameron Jerome, Ricardo Fuller and Glenn Whelan and within two minutes, the Potters were level.
Jerome netted the equaliser after powering through Phil Jagielka and John Heitinga, and the much-improved home side might have gone on to win it from there.
Pulis was reluctant to take too much praise for the second-half intervention he said he had made simply because his team, playing the 54th game of a gruelling season, were looking so weary.
Referring to the substitutions, Pulis - whose side moved up a place to 13th - said: "Sometimes you get them right, sometimes you don't.
"I'm standing here and I'm no genius - you don't know until they come on what effect they will have, and if anybody tells you any different, well then I'm telling you the truth.
"I just felt Crouchy and one or two of them looked tired."