An anti-tank rocket launched by Hamas fighters killed the Israeli soldier, Hamas officials said. One other Israeli soldier was reportedly injured.
The Israeli army did not immediately have any comment on casualties.
"The question is will Fatah be strong enough to contain such a powerful 'partner' or will it be just exploited and eroded over time?
Send us your views
However, an Israeli army spokeswoman confirmed that troops were operating in the area and that an air raid had been called in against a "cell of gunmen who opened fire against IDF [Israeli army] forces operating in Beit Hanoun area".
The third Palestinian, shot dead near the Sufa crossing with Israel on Monday, was Ahmed Abu Tahun, also from the armed wing of the movement, Hamas officials said.
An Israeli army spokesman said that Israeli forces "operating against terror infrastructure" had reported hitting at least two Palestinians in exchanges of fire in the area.
On Sunday, Israel began restricting fuel shipments to the increasingly isolated territory in a campaign defence officials hope will put pressure on fighters to stop rocket attacks.