Leading 7-1 from the first leg, City picked up where they left off in Switzerland – Beckie lifting over the keeper after a pass from Keira Walsh.
The Canada striker got on the end of a low Walsh pass to make it 2-0 and later lashed in to complete her treble.
Sub Pauline Bremer rounded the keeper to wrap up an 11-1 aggregate victory.
Last season’s SWISs league runners-up offered little resistance, with City manager Nick Cushing able to take off Walsh and England skipper Steph Houghton on the 55-minute mark, shortly after Beckie’s third goal.
The result means City better their efforts in the competition last season, when they were knocked out in the last 32 by Atletico Madrid.
Elsewhere, England winger Nikita Parris scored twice as holders Lyon thrashed Russian side Ryazan 7-0 to complete a 16-0 aggregate victory.
Both City and Lyon will be seeded in Monday’s last-16 draw, with the games to be played on the 16-17 and 30-31 October.