The Cormorant Alpha platform and its pipeline infrastructure were also shut down as a precaution following the discovery of the leak at 9.40am.
Seventy-one of the 145 staff members on board were removed from the rig, operator TAQA Bratani said.
The leak, discovered in one of the rig's legs during maintenance work, has now been contained, a company spokesman confirmed. All members of staff are said to be safe and well.
The incident marks the second time this year that the platform had to be partially evacuated following the discovery of a leak.
In January, the pipeline system was stopped and 92 workers were airlifted from the Cormorant Alpha rig after a similar leak was found in a platform leg.
The spokesman said "non-essential" workers were taken off the platform on standard scheduled flights.
He said: "TAQA Bratani can confirm that a hydrocarbon release detected in one of the Cormorant Alpha platform legs has now been contained, with no further hydrocarbon release.
"The downman of 71 non-essential personnel has been completed via the crew change flights. Everyone is safe and well.
"The hydrocarbons discovered at 0940 today during maintenance work have now been flushed through with sea water.
"The platform and all pipeline infrastructure remains shut down as a precaution.
"The release was contained within the leg and no hydrocarbons were released into the environment.
"TAQA continues to monitor the situation on Cormorant Alpha and is working with its partners to have the Brent pipeline system operational as soon as possible."
The installation, owned and operated by TAQA Bratani Ltd, is situated 232 miles (373km) from Peterhead and 94 miles (151km) from Lerwick on Shetland.
The company said its priority is the safety and well-being of the people on board the facility.
Environmentalists voiced concerns following the latest incident.
WWF Scotland director Lang Banks said: "While it's a relief to hear staff are safe and the leak has been contained, serious questions must be asked about the integrity of this ageing rig.
"In the interests of the environment and all oil workers it would be wise for the relevant authorities to take a much closer look at this platform, and those of a similar age dotted throughout the North Sea."
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