The group, calling itself the Qaeda Organisation of the State of Palestine, also vowed to target other senior officials, including Mahmoud Abbas, the Palestinian president.
"Your brothers in the Qaeda Organisation of the State of Palestine were capable of reaching the workplace of ... [Tareq] Abu Rajab," said the statement on Sunday, which was not posted on websites usually used by groups such as al-Qaeda.
"The mujahidin were able to plant a bomb in the elevator," it added.
The statement's authenticity could not be immediately verified, but an Amman-based journalist told Aljazeera that the statement could not be taken seriously.
Abu Rajab, the Palestinian general intelligence chief and Abbas ally, was seriously wounded after an explosion in an elevator at his Gaza Strip headquarters that killed one of his bodyguards and wounded five others.
The statement said Abu Rajab survived because the explosives detonated before the elevator door had closed.
"The model for jihad and martyrdom has been well exemplified by Hamas and the Islamic Jihad ... it is unlikely that al-Qaeda could have a real existence in Palestine"
The group also vowed to target Abbas and Mohammed Dahlan, a former security chief.
Both men are senior figures in the Fatah group which was in government until the landslide victory of Hamas in January elections.
Yasir al-Zaatra, a journalist in Jordan, told Aljazeera on Sunday that the statement contained "some of the language of Hizb al-Tahrir, some al-Qaeda language as well as Hamas terminology.
"Thus we cannot take it seriously."
Al-Zaatra said it was unlikely that al-Qaeda was active in the occupied Palestinian territories.
"Al-Qaeda is not a company with subsidiary branches with a line organisation," he said.
"There is a model that proliferates in several parts of the world. Such a model could be available in Palestine as it looks politically attractive."
But, he added, "the model for jihad and martyrdom has been well exemplified by Hamas and the Islamic Jihad. As long as this model is prevailing, it is unlikely that al-Qaeda could have a real existence in Palestine".
Israeli officials have warned of al-Qaeda agents infiltrating Gaza, which it withdrew from last year. Previous such Israeli claims have been rejected by the Palestinians.