The incidents followed a three-day lull since an upsurge in violence last week in which Israel killed at least four Gaza militants as dozens of rockets were fired at Israeli towns, damaging some homes and wounding several agricultural workers.
An Israeli air strike before dawn on Sunday struck two gunmen from the Palestinian enclave's governing Hamas movement as they rode a motorcycle near the central town of Khan Younis, local officials said. One man was killed and the other wounded.
An Israeli military spokesman said the air force had targeted a squad preparing to fire a rocket into Israel.
Hamas said its gunmen had fired mortar rounds at Israeli ground forces who had penetrated the coastal territory nearby. The military said those soldiers, who were unhurt, had been carrying out "routine work along the boundary fence."
Separately, two Palestinian rockets fired from Gaza struck Beersheba, a city 25 miles away, causing no damage, the military spokesman said. Beersheba sounded air raid sirens and shuttered its schools as a precaution against further attacks.
The Popular Resistance Committees (PRC), one of several smaller Palestinian factions in Gaza that often operate independently of Hamas, said it had launched one of the Beersheba rockets. There was no immediate claim for the second.
Though Islamist Hamas is hostile to the Jewish state, it has recently sought to avoid cross-border confrontations as it tries to shore up its rule of Gaza in the face of more radical challengers and to build relations with potential allies abroad.
Israel's policy is to hold Hamas responsible for any attack emanating from Gaza.
* Reporting by Nidal al-Mughrabi; Writing by Allyn Fisher-Ilan; Editing by Peter Cooney and Andrew Osborn