Puntland's authorities said they also rescued a second vessel on Tuesday that had been hijacked in May.
The deaths aboard the Palau-flagged ship appear to be the first at the hands of Somali pirates in over a decade.
The ship with 17 crew members was seized by pirates in April during a resurgence of Somali piracy that experts have attributed, in part, to impacts of the Iran war, including elevated oil prices that make tankers more attractive targets.
Puntland's information ministry said in a statement that a rescue mission had successfully freed the ship and captured 16 pirates.
"Before the operation begun, the pirates on board the ship killed five of the crew. Among those killed were three from Pakistan, one from Myanmar and one from India," it said.
Four crew members - three Pakistanis and one Indian - were also injured, the statement said.
The rescue follows another rescue mission last week, when Puntland's maritime police force freed the US-sanctioned tanker Sibu 1, which Somali pirates had hijacked in August.
Puntland's Maritime Police Force (PMPF) said in a statement late on Tuesday that it had also rescued an oil tanker called MT Eureka which had been hijacked in May after its hijackers fled when police approached the vessel.
"As PMPF forces approached the area, the pirates fled towards the shore and moved inland," it said.
"PMPF forces continue to pursue the individuals involved and are conducting further operations to apprehend those who escaped."
The two rescues bring the total to five out of the six ships held by pirates off the Somali coast as of late August that have now been freed.
A previous wave of piracy from 2008 to 2014 disrupted one of the world's busiest shipping routes, cost the global economy billions of dollars and resulted in hundreds of millions of dollars in ransom payments.
(Reporting by Abdi Sheikh and Abdiqani Hassan; Writing by Vincent Mumo Nzilani and Aaron Ross; Editing by Peter Graff, Alistair Bell and Thomas Derpinghaus)

A captured suspected Somali pirate smiles during an introduction to the media in Mumbai, India, 23 March 2024. The Indian Navy's warship INS Kolkata arrived in Mumbai after an operation in the Arabian Sea where it captured 35 suspected Somali pirates. The suspects were handed over to Mumbai police for legal action in accordance to Indian laws, specifically the 'Maritime Anti-Piracy Act 2022'. EPA/DIVYAKANT SOLANKI
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