The two technically skilled European sides largely cancelled each other out in the sweltering heat for the first hour and sixth-ranked England had to wait until the 76th minute to take the lead, Bronze thumping home a shot from 22 yards out.
England, who have now reached the quarter final for the third successive World Cup, will meet Canada on Saturday in Vancouver and fancy their chances against the home team who have largely failed to shine.
"We have an excellent record against this Canada team ... the momentum we got from this game will put us in a great place going into the quarter final," said upbeat England coach Mark Sampson.
With temperatures on the pitch hovering around 40 Celsius (104 Fahrenheit) at the start, the players initially adopted a slow pace. Norway had two good chances in the first 45 minutes but could not beat England keeper Karen Bardsley.
Eleventh-ranked Norway, the 1995 World Cup champions, started to lose their dominance when Sampson took off tricky, diminutive midfielder Fran Kirby in the 54th minute and replaced her with the more imposing Jill Scott.
"I felt we needed to get a little tighter in midfield and gain more territory," said Sampson.
Seconds after the switch, though, Solveig Gulbrandsen broke the deadlock, getting her head to a corner at the near post and flicking the ball in off the bar.
But England started to look much the better team and deservedly drew level from a corner in the 61st minute when captain Steph Houghton outmuscled two defenders to score with a fine header of her own.
England have now won three in a row at this World Cup after losing their first game.
Norwegian coach Even Pellerud said his team had started to struggle with the pressure after half-time, especially in the wake of Houghton's goal.
"We should have finished the game in the first half and we didn't," he said, congratulating Sampson for the way he had changed England's tactics.
(Editing by Frank Pingue / Ian Ransom)
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