During the standoff, Israeli forces shot into the crowd to "make the rioters back off", the official said.
Before the protest, the Israeli army posted signs telling Palestinians that they faced "Danger of Death" if they tried to approach the Sufa crossing.
The demonstrators, who marched from the cities of Rafah and Khan Younis in southern Gaza, burnt tyres, waved green Hamas flags and chanted "Dismantle the siege".
In June, Hamas took control of the Gaza Strip from forces loyal to Mahmoud Abbas, the Palestinian president.
Israel has since closed all crossings, allowing only humanitarian aid and Palestinians with urgent medical needs to cross.
In recent months Hamas has mounted several protests demanding an end to an Israeli-led economic blockade.
Separately, Palestinian medical workers said a 65-year-old woman died on Friday from wounds suffered a day earlier during an Israeli army raid in the southern Gaza Strip.
Dr Moaiya Hassanain said the woman was near her home around 275m from the border fence when she was shot on Thursday evening.
The Israeli military says it has no record of any shooting in the area at the time.