Bronze fired into the top corner from the edge of the penalty area as England battled against Norway.
Solveig Gulbrandsen's opener had given the Norwegians a lead, but Steph Houghton equalised with a header from a corner before Bronze's winner.
At the end of the match the substitutes ran on to the pitch to join their team-mates in celebrating the victory.
It is the first time England have won a knockout match at the Women's World Cup.
England now have the opportunity to better the quarter-final defeats suffered in each of their three previous World Cups.
They'll play Canada in Vancouver on Sunday.
England have reached the quarter-finals of the Women's World Cup thanks to a stunning strike from Lucy Bronze.