The projects - a transmission line for exporting 500 MW from West Bengal and a 1,320-MW thermal power project in Bangladesh - were inaugurated by Singh and his counterpart Sheikh Hasina through video conferencing.
The Maitri thermal power project is being developed by Bangladesh-India Friendship Power Company, a joint venture between NTPC and Bangladesh Power Development Board.
Inaugurating the projects, Singh said it was a "historic moment when we embark on a new partnership for prosperity between our two countries".
Observing that the destinies of the two countries are "inter-linked", he said, "The initiatives being undertaken today strengthen the bonds of friendship between India and Bangladesh and add a rich new dimension to our bilateral relations."
Sheikh Hasina, present at a sub-station in western Bheramara, 240 km from Dhaka, was joined by India's New and Renewable Energy Minister Farooq Abdullah.
"The inter-grid connectivity is part of an immediate solution which would go a long way to alleviate the power deficit in Bangladesh," Hasina said. Such cooperation will allow the two sides to embark on more ambitious projects, she added.
The formal opening of the transmission line came a week after India launched a test transmission of electricity. The inauguration began with the supply of 175 MW to Bangladesh's National Grid.
Bangladesh will import 250 MW from the Indian government's "unallocated quota" and another 250 MW will be supplied by a private firm.
However, groundbreaking for Bangladesh's biggest-ever joint venture coal-run power project came against the backdrop of protests by environmentalists, who fear the venture near the Sundarbans, also shared by India, will endanger the world's largest mangrove forest.