Martin's position is now in serious peril just three months after being hired after failing to win any of Rangers' opening five league games for the first time since 1978.
The Glasgow giants trail Hearts by nine points and could fall the same number behind fierce rivals Celtic should the Scottish champions win at Kilmarnock on Sunday.
Furious Rangers supporters vented their frustration throughout the 90 minutes with their patience for the former Southampton boss already exhausted.
To rub salt into the home side's wounds, it was a long-time Rangers target in Lawrence Shankland who did the damage for Hearts.
Shankland spun onto Claudio Braga's pass to fire the visitors into a deserved lead on 21 minutes.
Rangers showed flashes of sparking into life as Thelo Aasgaard's effort came back off the crossbar.
However, their first league defeat of the season was sealed when Mohammed Diomande pulled down Harry Milne inside the box eight minutes from time.
Shankland's penalty was saved by Jack Butland but the Scotland international was first to the loose ball and slotted in.