John Stones also scored twice while Jesse Lingard sent in a fine curler, and Kane gave credit to his teammates rather than celebrating the first World Cup hat trick for an England player since Gary Lineker did it in 1986.
The Tottenham striker bagged his third after a 61st minute shot by Ruben Loftus-Cheek hit his foot and bounced in.
The 24-year-old English captain became a national hero pretty quick at this World Cup after scoring two goals - including a attractive victor in stoppage time - in England's opening 2-1 win over Tunisia last week.
Largely solid at the back, although there were a couple of lapses stronger teams would have punished, and industrious in midfield, England found the scoring touch, and it will have pleased Southgate that the goals came from multiple sources. "It's been really good, we've been taken care of in Russian Federation", he added.
But in truth, it was a humbling experience for Panama against a side known to get nervous and slip up against less illustrious rivals on the big stage.
"Panama is like a small young child in football".
However much they tried, Panama struggled to block England's route to goal. That's important in the ethic of the team we're trying to create.
Kane and Fontaine are two of just three players to have achieved this feat - Hungary's Sandor Kocsis scored seven goals in the first two matches for his team at the 1954 World Cup.
In other action, Japan and Senegal will go head-to-head in Group H action after they both picked up victories in their opening games.
Though limited in ability, Panama's forwards did get into space behind England several times, showing that even if they do have Kane up front, plenty of work is still needed in defence.