United, kicking off their league campaign without Sir Alex Ferguson in charge for the first time since 1986, were handed a tricky opening five matches in fixture lists published on Wednesday.
They host Chelsea, with Jose Mourinho back in charge for the second time, on August 24 before Moyes makes a return to Merseyside with a game at Anfield against old foes Liverpool on August 31.
The champions then take on promoted Crystal Palace before making the short journey to the Etihad Stadium for the first Manchester derby against last year's runners-up with their new Chilean manager on September 22.
"Lively start to the season! Let us at them," declared United defender Rio Ferdinand on Twitter.
Everton, with Roberto Martinez stepping into Moyes's shoes, start out at Norwich City before hosting Chelsea as their fourth match of the season. They visit Manchester United on December 3.
Liverpool start the season at home to Stoke City, now under the leadership of Mark Hughes, while Arsene Wenger's Arsenal are at home to Aston Villa at Emirates Stadium.
