The win secured England a clash with Norway but the Lionesses’ 3-0 last-16 victory was overshadowed by a remarkable twin VAR protest from the Cameroon players.
After skipper Houghton had fired England ahead with a fine 14th minute free-kick, the game was then marred by farcical scenes on 45 minutes after a VAR review adjudged Ellen White was onside in scoring the Lionesses' second goal after the strike was originally disallowed.
After referee Liang Qin made her decision, the entire Cameroon squad gathered in a huddle appearing to question the decision and initially refusing to play on with the game held up for almost six minutes.
And there were more extraordinary scenes early in the second half after a VAR review had ruled out a Cameroon strike by Ajara Nchout for offside after the effort originally been awarded.
The decision prompted another mass Cameroon protest with Nchout in tears as the Cameroon players remonstrated with the officials before play eventually resumed after another lengthy delay.
England boss Phil Neville made three changes from the side which beat Japan in midweek, with five City players named in his starting line-up.
And there was a landmark 18th appearance at a World Cup for Jill Scott, which meant the City midfielder overtook Peter Shilton become the country’s most capped player at a World Cup.
England quickly settled into their rhythm enjoying the bulk of possession and territory with Cameroon content to try and break using their pace.
And after a cagey start, the Lionesses who struck first on 14 minutes thanks to a terrific free-kick from skipper Houghton.
Cameroon’s Augustine Ejangue was adjudged to have passed the back to her keeper just six yards out.
And after an interminable wait with Cameroon’s players lining up in unison on the goal line, Toni Duggan fed Houghton who showed great technique and vision in drilling the ball home in the bottom right hand corner to give England the perfect start.
England then doubled their advantage just before half-time through White, who continued her fine World Cup campaign by claiming her fourth goal of the tournament.
Lucy Bronze was the architect, playing a delicate through-ball to White who coolly applied a fine finish from 15 yards.
The effort was originally ruled out for offside before a VAR review overturned the decision and sparked a furious and prolonged protest from the Cameroon squad with the game delayed six minutes amidst farcical scenes.
Incredibly, there was yet more VAR controversy early after the restart – this time after a Cameroon goal was awarded then overturned for offside.
Nchout appeared to have rocked England by firing home past Bardsley but after checking with VAR official, referee Qin chalked off the effort for offside in the build-up.
The decision prompted another outpouring of protest and emotion from the Cameroon with Nchout in tears and needing to be consoled by her manager Alain Djeufma with another lengthy delay before play resumed.
To their credit, amidst the mayhem England retained their composure and after Bardsley had made two fine saves to further enhance her reputation, the Lionesses sealed victory with a third goal on the hour.
Alex Greenwood then skillfully steered home Duggan’s corner to create history as England scored three goals in a Women’s World Cup knockout game for the first time to book a quarter-final date against Norway in Le Havre on Thursday.
There was a late scare when Houghton was the victim of a poor challenge from Alexandra Takounda who received a yellow card after another VAR review but, fortunately the City and England skipper was able to continue.
Ultimately it was a game England would have been glad to get out of the way. Now the focus is all about the quarter-final.