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.
Many streets in the 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.
Hundreds of houses across the state were flooded, many trees toppled and roads submerged, disaster officials said.
Penang Chief Minister Lim Guan Eng said 1,968 people were evacuated in four districts, most of them on the mainland. The state is divided between the island and several districts on the mainland.
At least two people were killed -- one who drowned at home and a second in an old folks' home, state news agency Bernama said.
A hospital was also hit by the flooding, prompting the evacuation of over 100 patients and four babies.
"We remain fearful that there may still be untoward incidents because of the strong winds, the like of which have never experienced before, and flash floods might recur," Lim said.
Defence Minister Hishammuddin Hussein said the military had been deployed to help the flood victims.
"Let’s all pray that the situation in the flood-hit areas will return to normal soon," he said.
Britain's Prince Charles and his wife Camilla are due to visit Penang on Tuesday as part of a tour of Malaysia.