The League Two newcomers fell behind on 27 minutes when Joshua Mullin headed in a free-kick from Nicky Cadden.
Mullin then provided a neat cross for Daniel Mullen to knock in right on the stroke of half time.
Mikey Miller crashed a shot against the woodwork as the visitors from League One continued to dominate before Liam Buchanan made it 3-0 from close range.
Ousman See went close for the hosts but was denied a landmark goal by a smart save from Liam Kelly.
Livingston move up to second place in Group G, behind St Mirren, who have two wins from two.
Edinburgh City's first competitive match as an SPFL side ended in League Cup defeat to Livingston at Meadowbank.