The International Atomic Energy Agency (IAEA) has repeatedly expressed fears over the safety of the plant, which is Europe's largest atomic power station.
Russian forces took control of the six-reactor plant in embattled southern Ukraine in March last year.
"We are living on borrowed time when it comes to nuclear safety and security at the Zaporizhzhia Nuclear Power Plant," Grossi said in a statement.
"Unless we take action to protect the plant, our luck will sooner or later run out, with potentially severe consequences for human health and the environment," he added.
Two landmine explosions occurred outside the plant's perimeter fence -- the first on 8 April, and another four days later, according to the statement.
It was not immediately clear what caused the blasts, it said.
Grossi met senior Russian officials in Kalingrad last week and prior to this with Ukrainian President Volodymyr Zelensky in Zaporizhzhia to discuss a safety plan.
He also warned on Thursday that the plant continued to depend on a single still-functioning power line, posing "a major risk to nuclear safety and security".
A back-up power line damaged on March 1 has still not been repaired, the IAEA said.
It also said the staffing situation at the plant remained "complex and challenging", in part because of staff shortages.
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