Ian McCall's men had six different scorers - Ross Docherty, Craig McGuffie, Michael Moffat, Craig Moore, Andy Geggan and Declan McDaid.
Ryan Sinnamon scored a late consolation for Annan as Ayr made it four wins out of four in Group E.
Lee Erwin scored on his Kilmarnock debut as they beat Dumbarton 3-0 to qualify as one of the best runners-up.
Ayr will now be seeded for Sunday's draw, having scored 15 goals and conceded just three in an impressive start to the season, with 11 of those goals coming in their past two games.
Kilmarnock will join their Ayrshire rivals in the second round after Chris Burke and debutant Erwin both scored their first goals for the club in the win at home to Dumbarton.
Burke headed in Dom Thomas' cross eight minutes into the second half, and ex-Motherwell winger Thomas was again the provider as Kris Boyd doubled the lead with another header.
Erwin clipped in a third with a nick off the post 12 minutes from time.
Ayr United marched into the second round of the Scottish League Cup with a thumping 6-1 win over Annan Athletic.