The hosts looked battered and beaten after Ben Mee's headed opener just after half-time only for a crazy 379-second period to turn the match on its head as Michael Keane and new signings Andros Townsend and Demarai Gray all scored.
In his first two home matches Benitez has twice achieved what predecessors Marco Silva and Carlo Ancelotti struggled to do in winning league games from losing positions - with the club managing just two victories in 59 previous Premier League encounters when conceding the first goal.
Asked what he had done with effectively the same players, the Spaniard - who made a tactical switch from 3-4-3 to 4-3-3 immediately after his side equalised - said: "Not too much. You can see from the first minute of the first game and in every single game the commitment is there.
"They want to win and compete, they have the desire and want to do well. When you have a problem like you concede a goal they want to react, I think the fans want to react and fans give them more incentive they can do it.
"Every manager wants to win but when you win at home in front of your fans you have to enjoy it for sure.
"Hopefully the next one (game) we can score goals and it will be easier for my health."
Everton achieved the result without last season's leading goalscorer Dominic Calvert-Lewin, whose spell on the sidelines may extend beyond the initial two to three weeks Benitez first estimated.
"Could be longer. For me it will be more than two or three weeks," he added.
"He broke his toe (before the international break) and then he was playing with an injection and pain.
"It was not easy for him and he was not training like the rest of the team and that is why he had the problem."
Burnley boss Sean Dyche admitted the "madness of football" cost them dearly, leaving the Clarets still looking for their first win of the season.
"I was very pleased up to the (Everton) goal as we controlled a lot of the game away from home," he said.
"We conceded an awkward first and then after that (it was) the madness of football. We have a mad six minutes and the game has got away from you.
"That is something we have to get a grip of as it has happened twice this season with the Brighton game (where they conceded two in five minutes)."