There were fatalities in both countries after more than two million residents moved to shelters across Bangladesh's vast coastal region, according to officials and news reports.
At least seven people were killed in India's West Bengal state, where the storm first made landfall at around midnight on Saturday, the Press Trust of India news agency reported.
The storm then made its way to neighbouring Bangladesh, where seven people died, according to the United News of Bangladesh news agency.
Enamur Rahman, Bangladesh's junior disaster management minister, said about 5,000 homes across the country's coastal region had been damaged.
Trees were also uprooted by the cyclone and the agriculture ministry said more than 494,200 acres of crop land had been damaged in Bangladesh.
Packing winds of up to 75 miles per hour and gusts of up to 80mph, Bulbul weakened when it started crossing Bangladesh's southwestern coastal region, dumping heavy rain across the country.
Bangladesh's weather office said the cyclone first slammed ashore at Sagar Island in the southern part of India's West Bengal state before moving to Bangladesh.
Its path included the sou.thwestern Khulna region, which has the world's largest mangrove forest, the Sundarbans, which straddles the Bangladesh-India border.
Reporting by agencies.