Skipper Harry Kane claimed a hat-trick, including two penalties, with Jesse Lingard scoring the other goal as England joined Belgium at the top of Group G.
The two teams, who are level on goal difference and goals scored with England enjoying a better disciplinary record, now meet on Thursday to determine who wins the group.
It was England’s biggest-ever win at a World Cup finals and the Three Lions were set on their way by Stones who, having started the game alongside Club colleagues Raheem Sterling and Kyle Walker, put Gareth Southgate's side ahead with a towering eighth minute header.
In doing so, Stones became the first City player to score at the World Cup finals for England since Trevor Francis back in 1982.
After skipper Kane had doubled the lead from the penalty spot, Lingard added a fine third goal with a long-range effort before Stones struck again.
The Barnsely-born defender arrowed in a close-range header after Raheem Sterling’s close-range effort had been palmed out by the Panama keeper.
It was just reward for another superb all-round display by the 24-year old with both Sterling – who contributed two of England’s assists – and Walker also impressing again on the biggest stage of all.
And a dream first-half for England was sealed just before the break when Kane smashed home another penalty after being man-handled in the box.
In strength-sapping temperatures in Nizhny Novgorad, England eased off in the second half and after Kane had somewhat fortuitously sealed his hat-trick with a deflected effort off Ruben Loftus-Cheek’s shot, Southgate opted to freshen up the side.
Amongst the changes, Fabian Delph became the fourth City player to figure for the Three Lions when he came on off the bench.
Ironically the final word went to Panama who gained some consolation through Felipe Baloy but nothing could deflect from a day to remember for England.
Reflecting on his and England's dream display, Stones said it was a moment to cherish.
“It is great to get through, I remember watching four years ago and knowing how hard the lads had worked to get through. This time playing in this game it is a special moment and we wanted to do the country proud, get through to the next step and see where we can go," he said.
"To score my first goals for England was something special, especially at a World Cup. I’d have loved to have got a clean sheet, that is something to take forward into next games and hopefully I can get a few more goals form set pieces."