Videos posted by Indian newspapers -- apparently filmed by some of the passengers on the crowded ferry -- showed the open speedboat cruising through the waters before it turns sharply and rams the ferry.
Photographs later showed the ferry capsized.
"Doctors have declared 13 people dead", Maharashtra state chief minister Devendra Fadnavis told reporters, adding more than 100 people had been rescued.
"I pay my heartfelt tributes to those who died in this accident", Fadnavis said, adding that "we share the grief of their families".
The ferry was carrying passengers from the Gateway of India -- a colonial-era monument on the waterfront that is symbolic of Mumbai -- to the popular tourist spot of Elephanta Island, famous for its rock carvings.
"A Navy craft undergoing engine trials lost control and collided with a passenger ferry," India's navy said in a statement, adding that it regretted the "tragic loss" of lives.
The navy said there had been an "engine malfunction".
The 13 dead included one naval servicemember and two representatives of an equipment manufacturer taking part in the tests, the navy said.
Two people seriously injured were taken to the naval hospital.
"Search and rescue efforts were immediately launched", the navy said, with four naval helicopters flying overhead, while 11 naval craft and four coastguard or police boats searched the waters.