The army was deployed to provide help after a torrential downpour lasting for 15 hours sparked the worst inundations in the northern state for years.
Hundreds of houses across the state were flooded, many trees were toppled and roads cut off, disaster officials said.
Many streets in the state capital George Town on Penang Island, a UNESCO World Heritage site which attracts hordes of foreign and domestic tourists, were submerged in deep dirty water.
Officials said the floodwaters reached up to 12 feet (3.5 metres) and submerged houses in low-lying areas, with pictures showing cars almost entirely under water and victims being rescued in boats.
"Floodwater quickly inundated our house, destroying everything inside. This is the worst flood we have ever experienced," housewife Ramlah Awang, who was forced from her home in the town of Perai, told national news agency Bernama.
The five killed consisted of four elderly people who drowned and a Bangladeshi man crushed by a falling tree, the agency said.
Britain's Prince Charles and his wife Camilla are due to visit Penang on Tuesday as part of a tour of Malaysia.
Penang's fire and rescue department said a total of 3,365 people had been forced from their homes as of Sunday evening and taken to shelters.
The worst-hit area was Seberang Perai Tengah district on the mainland, with almost 2,000 people forced to flee.
The state is divided between Penang Island and several districts on the mainland.
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