"The Al-Qassam Brigades announce that the death penalty pronounced against its member Mahmud Eshtawi has been applied today at 1600 hours," Hamas's armed branch said in a statement.
Executions have previously been carried out in the Gaza Strip, including in public squares in the Palestinian territory, but it appeared to be the first time Al-Qassam itself had sentenced one of its own through a court martial and executed him.
The statement did not provide details on the accusations against him other than to say that "the Brigades' military and Islamic judicial committee issued the sentence because he violated rules and ethics."
Eshtawi's duties included overseeing tunnels that have previously been used to store weapons and carry out attacks against Israel, the sources said.
According to the sources, he was in charge of a large unit and was previously a close associate of Mohammed Deif, the Al-Qassam chief who has been a frequent target of Israeli assassination attempts.
The Palestinian Centre for Human Rights said in late December that nine death sentences had been issued in the Gaza Strip in 2015 and two in the occupied West Bank.
Since the start of 2016, four Gazans have been handed death sentences after being accused of spying for Israel.
The Gaza Strip has seen three wars with Israel since 2008.