Hamas militants then pursued the vehicle as it raced back to Israel, prompting Israeli air strikes that killed "a number of people".
The two sides then began an exchange of fire.
IDF tweeted Sunday that an IDF officer was killed and an additional IDF officer was "moderately injured" after fire broke out in a special forces' "operational activity".
"Several alarms have sounded in southern Israel". The Iron Dome missile defense system intercepted at lease two projectiles during the subsequent exchange of fire.
A senior official with Hamas, the group administering the Gaza Strip, said on Sunday that an Israeli special forces team infiltrated an area near the southern city of Khan Younis in a civilian vehicle.
Egypt, alongside United Nations special coordinator for the Middle East peace process Nikolay Mladenov, has recently played a key role in attempts to mediate a ceasefire between Israel and the armed groups in the Strip.
"This is happening at a delicate juncture in Hamas-Israeli relations", said Hugh Lovatt, a regional expert at the European Council on Foreign Relations.
Egyptian mediators have been working intensively to maintain calm, and also hope to bring about national reconciliation between the Hamas terror group, which seized Gaza by force in 2007, and the West Bank-based administration of Palestinian Authority President Mahmoud Abbas. It is the first installment of $90 million that Qatar has pledged to cover the salaries of thousands of unpaid Hamas employees for six months.
"Every action, without exception, has a price", Netanyahu said Sunday. "If you can't handle the price you cannot lead". "And I can handle the price". Hamas responded by lowering the intensity of Friday's border protest.
Palestinian sources said the senior militant killed was Sheikh Nur Barakeh, a commander of the Izzedine al-Qassam brigades, Hamas' military wing, in Khan Younis. One Israeli soldier has been killed.
Zionist Prime Minister Benjamin Netanyahu, who is visiting in Paris, held consultations on the topic while the Zionist War Minister Avigdor Lieberman is holding consultations at the military headquarters in Tel Aviv.
Even his erstwhile allies attacked him for indirectly doing business with a militant group that has fought three wars against Israel and is sworn to Israel's destruction.
While Netanyahu has sought to avoid a large-scale military offensive in Gaza, the comments from his rival ministers signal that he is under pressure domestically for being soft on Hamas.
Tal Russo, a retired IDF general, told Israel's Channel 10 that the fighting likely stemmed from an intelligence-gathering mission gone awry.
He is due back in Israel on Monday, but while he is gone, Culture Minister Miri Regev is acting prime minister.