Fulham manager Magath had called for patience ahead of Saturday's clash at Pride Park after the London club lost their first three games in the second tier following their relegation from the Premier League.
But the experienced German could be on borrowed time after watching his side capitulate in the second half as Steve McClaren's side ran riot to leave Fulham bottom of the table.
Derby reacted well to that setback and Craig Bryson put them back in front. Martin then gave them breathing space two minutes later and scored a fourth goal before Simon Dawkins added insult to injury two minutes from time.
Victory for Derby moves them up to seventh spot in the table, which their arch-rivals Nottingham Forest lead after thrashing Reading 4-0 at the City Ground.
Free-scoring Forest have made a dream start to Stuart Pearce's tenure and his new signings produced the goods once again at the expense of Nigel Adkins' side.
Michail Antonio scored twice against his former club, while big-money signing Britt Assombalonga and Matty Fryatt were also on target again as Reading were put to the sword.
Watford occupy second spot after Fernando Forestieri scored twice in a 4-1 thrashing of nine-man Leeds United and Wolves are fourth behind Norwich courtesy of a late Mark Hudson own-goal at Molineux that consigned Cardiff City to their first defeat.
Hull City target Jordan Rhodes opened the scoring in Blackburn Rovers' 3-2 victory over Bournemouth, while Blackpool are still without a point following their 1-0 defeat at Lancashire rivals Wigan Athletic.
In the other games to be played on Saturday, Rotherham United won 1-0 at Millwall, Brighton and Hove Albion beat Bolton Wanderers 2-1 and Birmingham City drew 1-1 at Brentford.
