The death toll is expected to rise as rescue operations continue. At least 53 people are still missing.
A navy warship returned to the city's harbour on Wednesday with some of the 186 survivors rescued so far.
Thirty-five people have also been rescued from another of the three other barges stranded due to cyclone Tauktae.
The cyclone weakened after it made landfall on Monday, but at least 19 people are confirmed as having died on land in the storm.
The barge was working on a project for India's Oil and Natural Gas Corporation (ONGC). It was carrying personnel deployed for offshore drilling. When the storm struck, the barge's anchors gave way and it began drifting.
"We're still looking out for people in the area. We should be optimistic. Right now the conditions have improved. Hopefully, the worst is behind us," Captain Sachin Sequeira, commanding officer of the INS Kochi warship, told ANI news agency.
"The barge was sinking, so I had to jump into the sea. I was in the sea for 11 hours. After that [the] navy rescued us," a survivor told ANI.
After receiving distress calls from the barge also called Papaa-305 (P305), the navy's ship sailed under difficult weather conditions.
"[The] storm was just passing west of Mumbai [city] as we arrived on scene we took charge of the situation," Captain Sequeira said.
"An intensive search and rescue operation including helicopter search operations are ongoing to locate and rescue the missing personnel," Afcons, the infrastructure development company which chartered the barge, said in a statement.