The Spaniards will now play the winner of Sunday's round of 16 game between the Netherlands and South Africa.
The 20th-ranked Swiss matched their best World Cup finish.
They also bowed out in the round of 16 in their only other World Cup appearance in 2015.
Spain finished second in Group C, beating Costa Rica and Zambia by a combined 8-0 before being run ragged in a surprise 4-0 loss to Japan on Monday.
Bonmati netted her second in the 36th minute with some exquisite footwork that sent two defenders and keeper Gaelle Thalman the wrong way.
Persons:
Switzerland's Lara Marti, Spain's Ona Batlle, David Rowland AUCKLAND, Aitana Bonmati, Bonmati, Alba, Laia Codina, Jenny Hermoso, Cata Coll, Jorge Vilda, Maria Rodriguez, Codina, Ona, Redondo, Gaelle, Hermoso, Gaelle Thalmann, Lori Ewing
Organizations:
Soccer Football, FIFA, Spain, La Roja, Swiss, C, Roja, Thomson
Locations:
New Zealand, Switzerland, Spain, Eden, Auckland , New Zealand, Spain's Ona Batlle REUTERS, Netherlands, South Africa, Costa Rica, Zambia, Japan, Coll, Swiss, Redondo, Hermoso, Sydney