Rokeya Khanam, 50, said to be a leader of ruling Awami League's women unit, hit a concrete wall hard while fleeing a school committee meeting in northwestern Pabna, private TV channels and witnesses said.
"She suddenly fell down from the chair as the tremors jolted the scene... The doctors declared her dead as she was rushed to the hospital," media reports quoted witnesses of the school in Pabna as saying.
At least 50 people were being treated at a nearly hospital following the stampede in Savar area.
In northern Mymensingh, at least eight students of a primary school were injured in another stampede while rushing out of their two-storey school building.
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Hundreds of people hurriedly came out of their houses and workplaces in major cities as the high-rise buildings swung during the tremor followed by two subsequent aftershocks.
"The fire service sealed a 10-storied building evacuating the people inside," a police official said.
Several other buildings were also tilted during the quakes in different cities.
Experts were already fearing a major quake to hit the country as five fault-lines run through the country.
According to an expert study, an earthquake of 7.5 magnitude might kill nearly 100,000 people pulling down some 70,000 vulnerable buildings in the Bangladesh capital Dhaka alone.
Nearly 700 people were killed after a powerful earthquake measuring 7.9 on Richter scale struck Nepal today flattening houses and buildings including the centuries-old Dharhara tower and triggering an avalanche on Mt Everest.