Mark Hughes' side were good value for their success at the Britannia Stadium on Wednesday and rarely looked in trouble against Roberto Martinez's men, who continue to struggle in domestic action after a five-game winless league streak yet have progressed well in the UEFA Europa League.
Moses struck in the first half when he out-jumped Phil Jagielka and John Stones to head past Tim Howard, with substitute Diouf wrapping up the points when he netted with seven minutes to go.
Everton's best chance fellow to second-half substitute Arouna Kone, but his shot was beaten away by Asmir Begovic.
Stoke were without midfielder Stephen Ireland, who suffered a gashed calf in the 1-0 win over Hull City on Saturday, while Peter Crouch's reward for coming off the bench to score the only goal in that game was a starting berth.
Everton midfielder Darron Gibson made his first Premier League start since May 2013 following his recovery from a knee injury and defender Leighton Baines failed a fitness test on a quad injury, so Luke Garbutt continued at left-back.
Garbutt set up Gibson for a long-range shot that was easy for Begovic to deal with before Gareth Barry picked up his 10th yellow card of the season, meaning a two-game ban, when he fouled Charlie Adam and was cautioned by referee Mark Clattenburg.
Stoke, already without Ryan Shawcross, lost fellow central defender Philipp Wollscheid to injury with 15 minutes gone.
With just under half an hour gone, Stoke went close to breaking the deadlock.
Moses cut in from the left and fired in a right-footed 25-yard shot that Howard kept out with clenched fists.
Stoke were on top and a minute later went ahead.
Adam fed Phil Bardsley on the right and his fine cross was headed into the top-left corner by Moses.
Steven Naismith almost worked an opening, but he was kept out by the Stoke defence and then Aaron Lennon saw his shot smothered by Begovic as the home side remained in control to the interval.
Romelu Lukaku ran clear early in the second half, but his shot was across goal and did not trouble Begovic.
Substitute Kone stung the hands of Begovic as Everton looked for an equaliser that they rarely looked like scoring, leaving Martinez to bemoan his side's continued poor Premier League form.
Diouf sealed the win when he shot home after fellow substitute Marko Arnautovic had shot against the post.
