The fire started at around 12:22 a.m. local time at the fifth-floor restaurant, called 1 Above in Kamala Mills, according to Mumbai police spokesperson Deepak Deoraj.
Images of the scene showed a raging fire overtaking the rooftop and billowing smoke in the night sky.
There are two restaurants on the building's top floor -- 1 Above and another establishment, Mojo's Bistro. Kamala Mills is a former industrial compound that in recent years has been refurbished and now houses restaurants, bars and offices in Mumbai's Lower Parel area.
Eleven of the 14 victims were female and the remaining three were males.
A case has been registered against 1 Above's manager and partners, Deoraj said.
Indian Prime Minister Narendra Modi tweeted about the fire: "Anguished by the fire in Mumbai. My thoughts are with the bereaved families in this hour of grief. I pray that those injured recover quickly."
Eyewitness: Flames spread in a 'matter of seconds'
One diner who was present when the fire started told CNN that she and her dining companion noticed a "small fire" in the corner of the restaurant, but she had assumed staff would bring it under control.
However, customer Sulbha KG Arora had underestimated the speed at which it would spread.
"Before anyone had time to react in a matter of seconds the flames had spread very rapidly and engulfed the entire rooftop," she said.
"There was no time for anyone to try and bring the fire under control, the only thing possible to do was to try and escape from there."
Arora said there were around 100 people in 1 Above when the flames spread, and in a "bit of a stampede" diners rushed towards the entrance.
She says she was pushed down in the melee. "By the time I tried to get up, the whole roof above me was coming down in flames," she said.
Arora and her companion, an old college friend, instead escaped to the kitchen where staff helped them locate and exit through a fire escape door.
"There were quite a few flights of stairs we had to run down and as we kept running we could literally feel the fire chasing us because we kept feeling the heat behind us," she said.
Correction: A previous version of this article referred to Sulbha KG Arora as "he."