Last season's treble winners had almost 50 attempts on goal, with Clare Shine scoring a hat-trick.
Aberdeen, Rangers and Spartans all netted five goals in disposing of Jeanfield Swifts, Hamilton and Stirling University, respectively.
Hibernian won 1-0 away to Celtic and Glasgow Girls beat Hutchison Vale 3-1.
The Forfar Farmington v Hearts and Buchan v Queen's Park matches were postponed due to frozen pitches.
SWPL League Cup, first round
Aberdeen 5-0 Jeanfield Swifts
Glasgow City 14-0 Inverness
Celtic 0-1 Hibernian
Rangers 5-1 Hamilton
Glasgow Girls 3-1 Hutchison Vale
Spartans 5-1 Stirling Uni (aet)
Glasgow Girls 3-1 Hutchison Vale
Forfar P-P Hearts
Buchan P-P Queen's Park
Glasgow City opened the new season in ominous style, demolishing Inverness 14-0 in the first round of the SWPL League Cup.