Solveig Gulbrandsen gave Norway the lead early in the second half before a Steph Houghton header and Bronze’s long-range strike secured a 2-1 win, setting up a quarter-final tie with hosts Canada on Saturday.
England forged the first opportunity of the game after a delightful ball from Fara Williams was met by Bronze, who crossed for Toni Duggan but the striker’s header was wide of the target.
Norway should have gone ahead minutes later when Herlovsen was clean through on goal, but Karen Bardsley rushed off her line to deny the Norway striker and keep the score goalless.
Kristine Minde then had an opportunity at the far post but the midfielder was unable to get on the end of a teasing cross from the left hand side.
Karen Carney looked like England’s biggest threat in the first half and the Birmingham midfielder was denied a hopeful penalty when she went down in the area on the half-hour mark.
England had Bardsley to thank for keeping the score level at the interval when she stayed on her feet to deny Ada Hegerberg after a sloppy back-pass from Bronze.
England started the second half brightly but Bardsley unnecessarily palmed a wayward Gulbrandsen shot out for a corner on 54 minutes and it led to the opening goal of the game.
The cross was swung in from the left and Gulbrandsen flicked a header which bounced off the underside of the crossbar and over the line to give Norway the lead.
But England were behind for only seven minutes and their equaliser also came from a corner. Williams swung the ball in from the right-hand side and Houghton rose to power her header past the goalkeeper and get England back on level terms.
Substitutes Jodie Taylor and Jill Scott were beginning to influence the game as both sets of players began to look a little weary as the game wore on, and both had a part to play in England’s winner.
The pair combined on England’s right flank in the 76th minute before the ball was squared to Bronze, who hit a thunderbolt into the top corner which Ingrid Hjelmseth got a hand to but was unable to stop.
Carney had a chance to make it three late on but this time Hjelmseth got a hand to it to deny the midfielder. It made no difference to the outcome though as England held on to set up a Saturday quarter-final with Canada in Vancouver.
Faye White’s player ratings: Bardsley 9, Bronze 8, Houghton 8, Bassett 8, Rafferty 7, Chapman 7, Moore 8, Williams 7, Duggan 7, Kirby 7, Carney 8, Scott 8, Taylor 7.
The United States also progressed to the quarter-finals with a 2-0 victory over Colombia in Edmonton.
The Americans took advantage of the sending off of Colombia goalkeeper Catalina Perez with second-half goals from Alex Morgan and Carli Loyd to book a last-eight clash with China in Ottawa on Friday.
Perez was red-carded for bringing down Morgan in the area two minutes after the break, but Abby Wambach missed the penalty, denying her a record-equalling 15th Women’s World Cup goal to match Brazil star Marta.
At the other end of the pitch, Hope Solo was rarely tested as she earned a US goalkeeping record 174th cap.