England, ranked sixth, had fallen behind to a 54th-minute goal for 1995 champions Norway off a Solveig Gulbrandsen header.
But second-half goals from Steph Houghton and Bronze gave England their first win in the knockout round of a Women's World Cup.
"Bronzie was outstanding today. She recovered from a couple of errors in the first half that nearly cost us and KB (goalkeeper Karen Bardsley) came to our rescue," said coach Mark Sampson.
England next play hosts Canada on Saturday in Vancouver.
"We're ready for a big quarter-final and determined to keep this journey going."
Sampson made four changes to his starting lineup with midfielder Katie Chapman, defenders Laura Bassett, Bronze and Claire Rafferty all coming on.
But in a drab first half, Norway had the most chances, with Bardsley keeping England's sheet clean, superbly denying Isabell Helovsen early as England struggled in the Ottawa heat.
- 'Nice tactical changes' -
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Back after the break, Norway broke the deadlock with Gulbrandsen, 34, heading in off a corner for a goal which was confirmed by goalline technology.
And Bronze added the second with 14 minutes to go following good work from substitutes Jill Scott and Jodie Taylor to fire a powerful long range shot from 25 metres into the back of the net past a diving Ingrid Hjelmseth in the Norway goal.
