Israeli news website Ynet posted a screen shot of what it said was the offending tweet.
"#WARNING: Possible nuclear leak in the region after 2 rockets hit Dimona nuclear facility" in the south of the country, it read.
The army told AFP that "several incorrect tweets were posted" on its English spokesperson's account (@IDFspokesperson) late Thursday, saying the account had been "compromised."
The account posted an apology to followers and had erased any trace of the hackers' false messages by Friday morning.
Internet hacking group the Syrian Electronic Army claimed responsibility, posting a screen shot of the army's Twitter page with at least two of its fake tweets.
"Always via @Official_SEA16 Long live #Palestine" read the other tweet, linking directly to the SEA's official account.
The hacking came as Israel's army gave regular updates on rocket fire from Gaza hitting southern Israel, after a brief uptick in the past week in cross-border exchanges, including retaliatory air strikes by Israel.
Israeli warplanes pounded targets inside Gaza on Thursday and militants fired at least 34 rockets into Israel, the army said.
Israel warned Hamas on Thursday to put a stop to rocket fire from its Gaza power base and sent army reinforcements to the border with the Palestinian enclave.
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