Garnham, who has poured in 20 goals in nine games this season, added his name to the score sheet soon after.
A clever touch from a floating pass down the left led to a jink to the middle, where a stinging shot earned a second opportunity that was not to be missed.
The fourth - Garnham's second - was soft; the result of a goalkeeping error following a skimming strike from outside the area.
Former Tauranga City United player Connor Irvine's diving header was close to the pick of the goals before the half, before Guy Privett, Garnham and Gomez ran up the score in the second stanza.
Garnham said the Auckland Football Federation's ruling had left the side wanting to prove a point this season in the race for promotion.
"We want to get to the Northern League, that's our motivation. If Tauranga win the league we automatically go up because they can't go up, but we don't want to win it like that," Garnham said.
"We're doing well, we're coming together."
The side only had a pair of reserves but did not need much in the way of reinforcements.
"People work, but today we probably didn't need it. It was alright, we did what we needed to do. Every team we play is a challenge. It doesn't matter who it is - everyone is out to beat Fury."
AFC Fury 8 (Lee Garnham 3, Federico Gomez 2, Connor Irvine, Guy Privett, own goal) Melville United 0