JERUSALEM, June 18 (Maayan Lubell/Reuters) - Militants who crossed into Israel from Egypt's Sinai Desert fired on Israelis building a barrier on the border on Monday, killing one worker, before soldiers shot dead two of the attackers, Israel's military said.

Israel later launched air strikes killing four Palestinians in the Gaza Strip, including two militants from the Islamic Jihad group on a motorcycle. Two other militants were killed while trying to fire a rocket, Israel said.

The Sinai attack, launched soon after the Muslim Brotherhood declared victory in Egypt's presidential election, raised Israeli concerns about lawlessness in the area since the fall of president Hosni Mubarak last year.

We can see a disturbing deterioration in Egypt's control of the Sinai's security, Israeli Defense Minister Ehud Barak said, commenting on the attack.

We are waiting for the election results. Whoever wins, we expect him to take responsibility over all of Egypt's international commitments, including the (1979) peace treaty with Israel and security arrangements in the Sinai, and to put an end to these attacks swiftly, he told reporters.

Three gunmen crossed into Israel from the Sinai Desert, the Israeli military said.

A terrorist squad opened fire and possibly also fired an anti-tank rocket at an area where (Israel) is constructing the border fence, spokesman Yoav Mordechai said.

Soldiers who rushed to the scene killed two of the militants but could not find the third, who may have returned to Egypt, the military said.

A military source said the dead worker was an Arab citizen of Israel. There was no immediate claim of responsibility for the attack, which took place about 30 km (18 miles) from the Gaza Strip.

Israel is building a fence along the frontier to curb an influx of African migrants and boost security, and hopes to complete it by the end of the year. It will run along most of the 266 km (165 miles) from Eilat, on the Red Sea, to the Gaza Strip.

In August last year, militants crossed over the Egyptian border and killed eight Israelis, in the most serious attack in the area since the Egyptian popular uprising.

On Saturday, at least two rockets were fired deep into southern Israel, causing no damage or casualties. It was not clear whether they had been launched from Sinai.

Palestinian militants in the Gaza Strip, ruled by the Islamist group Hamas, have launched rockets at Israel from the coastal territory in the past. Israel says Palestinian militants have also crossed into Sinai to launch similar attacks.

Late on Sunday, Israeli aircraft carried out a series of strikes in the Gaza Strip in response to rocket fire from the enclave. Medical sources in Gaza said seven people were wounded. (Additional reporting by Saleh Salem in Gaza; Writing by Allyn Fisher-Ilan and Maayan Lubell; Editing by Kevin Liffey)