Switzerland kept three clean sheets to top Group A, but it took a mere five minutes for midfielder Bonmati to breach their defence for La Roja’s first goal.
Additional goals from her, Alba Redondo, Laia Codina, and Jenny Hermoso completed the rout to the delight of the crowd of 43,217 – a record for a football match in New Zealand, women or men.
Codina’s 45th-minute effort was a moment of redemption for the defender, who had earlier scored a calamitous own goal with a pass back from 40 yards that sailed past shocked goalkeeper Cata Coll.
The Spaniards will now play the winner of tomorrow’s round of 16 match 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.
Coach Jorge Vilda made five changes to his team from that loss for today, including dropping goalkeeper Maria Rodriguez in favour of Coll.
Bonmati got Spain on the scoreboard when she took a terrific turn with the ball to finish from close range but the Swiss equalised five minutes later when Codina put the ball through her own net.
It took La Roja – the Red One – six minutes to regain the lead when Ona Battle swung in a cross that Redondo met with a header from the back post into the far corner.
Bonmati netted her second in the 36th minute with some exquisite footwork that sent two defenders and goalkeeper Gaelle Thalman the wrong way. She knocked the ball to her left foot and calmly curled a shot into the empty net.
Codina stabbed the ball home during a goalmouth scramble before streaking away with outstretched arms and a wide smile of relief to celebrate a goal that sent Spain into halftime with a 4-1 lead.
Hermoso scored the only goal of the second half in the 70th minute, pouncing on a poor clearance by the Swiss that left her one-on-one with goalkeeper Gaelle Thalmann.
Bonmati, Redondo, and Hermoso all now have three goals in the tournament.