Jodie Taylor broke a 14-month goal drought to shoot England past Argentina and into the knockout stages of the tournament, and they now need a point against Japan on Wednesday to ensure they top the group.
It wasn't a win that came easy thanks to goalkeeper Vanina Correa, who pushed Nikita Parris' 28th-minute penalty off the post and then pulled off a super stop to deny the excellent Beth Mead four minutes before the break.
"I thought they were outstanding, we asked them to play with patience, control, rhythm, with speed, and we dominated the whole game like we did against Scotland apart from 11 minutes," said Neville.
England travel to Nice with their destiny in their hands after an emotional night for Fran Kirby, who lost her mother to a sudden brain haemorrhage when she was just 14, and Carly Telford.
Friday would have been her mother's birthday. Goalkeeper Telford, who played her first minutes in a major tournament on Friday, lost her mother to cancer last year.
"We're a family. We were aware today that it was Fran's mum's birthday. It's been well-documented the love and affection she had from her mother and how much she misses her," Neville said.
"We hope their two mums would be looking down on them and be very proud of their performances."
England topping the group would mean they should avoid a next-round clash with European champions the Netherlands, who continue their campaign against Cameroon on Saturday.
The fancied Dutch only just squeezed past New Zealand, who take on Canada in Grenoble, in their opening match thanks to a last-gasp Jill Roord header, and now face a tricky test against a Cameroon side they have never played before.
- Italy dreaming -
They will be hoping for a much-improved performance if they are to become part of a powerful-looking knockout line-up alongside England and Italy, who started the tournament as outsiders but have surprised everyone by qualifying from Group C after scoring seven goals in their opening two matches.
They are through thanks to Cristiana Girelli's hat-trick and a late brace from Aurora Galli, which also gave Milena Bertolini's side a perfect six points and put them top of their group.
Italy can now guarantee top spot by getting a result from their final game against Brazil, although their superior goal difference means they can lose and still maintain first place.
"I couldn't have even dreamed of scoring a hat-trick in the World Cup, it's an incredible personal satisfaction," said Girelli.
However the penalty with which Girelli opened her tournament account caused controversy, with referee Anna-Marie Keighley pointing to the spot after a VAR check in the 10th minute for a what looked like a soft foul on Barbara Bonansea.
Girelli then missed the spot-kick, but Keighley allowed her to retake following another VAR check that Jamaica goalkeeper Sydney Schneider had encroached.
The 29-year-old made no mistake the second time round -- two minutes after the original penalty was given -- before doubling Italy's lead 13 minutes later when she bundled home Bonansea's flick-on from a corner.
Girelli became only the second Italian woman to score a World Cup hat-trick just after the break when she beat Schneider to a looping cross and have the solid Azzurre dreaming of going deep in their first World Cup in two decades.