The 6.1-magnitude quake struck inland at 0737 GMT at a depth of just 10 kilometres (6.2 miles) in the mountainous Bener Meriah district in the heart of Aceh, the US Geological Survey said.
It destroyed houses in the district, some 320 kilometres (200 miles) from the provincial capital Banda Aceh, and triggered several landslides in the area.
Police and military personnel were deployed to lead rescue efforts, as pictures showed the walls of houses reduced to rubble and roads badly damaged and blocked by land slips.
"A man was working at a coffee plantation with his wife when a landslide caused by the quake struck," said the official, who like many Indonesians goes by one name.
"His body was found under a pile of earth soon afterwards," he said, adding that his wife and another woman were missing at the site.
Ema Suryani, a doctor at a health clinic in Lampahan city in the district, told AFP that a child died when the quake caused a wall to collapse.
"There are around 30 people seriously injured, some with head injuries. The rest have only light injuries like minor cuts and grazes."
Injured people had been transported from several affected villages in two trucks, she said.
The disaster agency official Fauzi told AFP that a man also died at a different hospital after being treated for injuries sustained in the quake.
People ran outside in panic in Banda Aceh as the quake shook houses, and in Medan city to the south of Aceh province on Sumatra island.
Aceh, on the northern tip of Sumatra, is regularly hit by quakes. In 2004 a massive tremor sparked a tsunami that killed 170,000 people in the province and tens of thousands more in countries around the Indian Ocean.
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