A second soldier was moderately wounded in the incident, near Mount Arif, said army spokeswoman Avital Leibovich.
Lt Col Leibovich said the militants had opened fire on the soldiers after crossing the border from Sinai.
The troops were guarding a site where work is continuing on a new security barrier along the border with Egypt.
The violence comes amid a crackdown by Egypt on Islamist militants operating in the Sinai Peninsula, launched after an attack last month left 16 Egyptian soldiers dead.
In that incident, the militants subsequently crossed the border with Israel in an apparent attempt to carry out another attack. However, they were killed in Israeli air strikes.
In June, militants fired an anti-tank missile at a vehicle convoy carrying construction workers helping to build the new border fence. One Israeli construction worker and two of the attackers were killed.
Analysts say that the security situation in Sinai has deteriorated following the fall of Egyptian President Hosni Mubarak last year, and that local Bedouin criminal gangs, radical Palestinian groups and possibly militant Islamists sympathetic to al-Qaeda have gained a foothold.
By Kevin ConnollyBBC News, Jerusalem
Israel's border with Egypt in the baking, hostile wilderness of the Sinai desert was among the most dangerous flashpoints in the Middle East until the two countries signed a peace treaty in 1979. They had been to war four times there in the previous 30 years.
Under Anwar Sadat and then Hosni Mubarak, Egypt kept an iron grip on the region offering Israel such strategic certainty that the border between the two countries was not properly fenced.
Since the fall of Mubarak concern is deepening in Israel that Egypt is less able, or less willing to maintain the same degree of control. There have been a series of incursions over the last year - in the most serious, militants killed 16 Egyptian soldiers before blasting their way through the border fence.
Israel fears more - and is building a fence which will run the entire length of the frontier. This latest attack came at one of the last points where work is still to be completed.
Lt Col Leibovich said the militants who crossed the border from the Sinai peninsula on Friday were heavily armed with rifles and equipped with bullet-proof vests and explosive belts.
The attack happened about half-way down Israel's 240km (150-mile) southern border, at a point where work is under way to complete the new barrier.
"[The militants] opened fire towards IDF troops that were guarding the workers in that area. Another force that was nearby... rushed to the area and targeted those three terrorists," Lt Col Leibovich told reporters.
"A very big terror attack was thwarted by the quick response of these soldiers," she added.
The dead Israeli soldier, named as 20-year-old Cpl Netanel Yahalomi, was reportedly shot in the head during the gun battle.
Egyptian security sources said one of the militants died when his explosive belt was detonated, while the other two were shot dead by the Israeli soldiers. The nationalities of the assailants were not immediately clear.
Defence officials said work on the border fence in the Mount Arif area was due to conclude in early November. Construction had been delayed because it was difficult in a remote, mountainous region, they added.
Israel's options in responding to Friday's attack are constrained, reports BBC defence correspondent Jonathan Marcus.
In the past it has chosen to attack Palestinian militants in the Gaza Strip, prompting rocket fire on Israel in return. But an Israeli strike into Egyptian territory is almost certainly ruled out; such a step could send already delicate Israel-Egypt ties into a downward spiral, our correspondent says.
Now both Israel and the US will be urging the Egyptian authorities to devote even more attention to security in Sinai, he adds.
An Israeli soldier and three militants have been killed in a clash inside Israel near its southern border with Egypt, the Israeli military has said.