Earlier, Australia beat Scotland 14-12 and the United States pulled away from Japan 30-17, leaving the two beaten teams on the brink of elimination.
On a day of tense matches in Whangarei, it was the Pool C showdown between the two Six Nations heavyweights that carried the most significance, with England and France strong contenders to meet again in the final next month.
Emily Scarratt scored all the points for England, including their lone try, as they extended their unbeaten run to 27 matches.
Looking in trouble at 13-0 down with 15 minutes remaining, France scored against the run of play through replacement forward Gaelle Hermet to give them a glimmer of hope.
However, England’s domination of possession and territory continued over the closing stages, although they continued to squander scoring opportunities against a staunch French defensive line.
England captain Sarah Hunter was not surprised by tight nature of the game.
“We know encounters between us and France over the last few matches have come down to fine margins. Tonight was no different, it’s two very good sides,” said Hunter, who equalled England’s record for Test caps with her 137th appearance.
The world’s top-ranked side forced fourth-ranked France to attempt more than 140 tackles in the first half, and that pressure told when outside centre Scarratt crossed.
Her conversion, followed by penalty goals on either side of half-time, helped set up victory against opponents stung by injuries to two key players in the first 20 minutes.
Scrum-half Laure Sansus and No.8 Romane Menager both had to be helped off the field and their fitness for the remainder of France’s campaign was unclear.
The two earlier matches involved four teams beaten in the opening round last week.
Scotland must now upset hosts New Zealand in the final round next week to have any chance of qualifying for the quarter-finals from Pool A, while Japan will face the same scenario against Pool B rivals Italy.
The little-fancied Japanese threatened a major upset when they led the US 5-3 at half-time.
They were still two points ahead with 25 minutes remaining but could not hold off the Eagles, who created tries for Joanne Kitlinski, Elizabeth Cairns, Jennine Detiveaux and Alev Kelter.
US captain Kate Zachary said her team reached another level when they realised what was at stake after a modest first-half display.
“Deep down inside, the players know how important that second half was,” Zachary said.
“We said we needed to shine in that second half and every player did.”
The match was a sharp contrast to the only other meeting of the two nations, when the US won 121-0 in pool play at the 1994 World Cup.
Earlier, Australia held their nerve after losing both of their hookers to red cards in the closing stages against Scotland.
Second-half tries to Bienne Terita and Ashley Marsters helped the Wallaroos overcome a 12-0 deficit at the interval but the match finished in tense circumstances when Marsters and Adiana Talakai were both sent off for high tackles.
Australia captain Shannon Parry was relieved following their first-up 41-17 loss to New Zealand.
“A bit of a nerve-wracking one, it was a real tough battle,” Parry said.
“We didn’t play as clinically as we wanted to but we’ll take the points, we’re still in this tournament and that’s what matters.”
It was a second narrow defeat for the Scots, who lost in the final minute, 18-15, to Wales last week.
