Fatah officials accused Hamas forces of opening fire from the nearby Islamic University, but Hamas said its men had come under attack from Fatah gunmen and shot back.
Organisers say that about 100,000 people had turned up at the commemoration event for the former Fatah leader.
Hamas had broken up some smaller demonstrations organised by its rival on Sunday, the third anniversary of Arafat's death.
Speaking at Monday's rally before the gunfire broke out, Odeh called the atmosphere in Gaza City "quite tense".
"Fatah promised everyone a show of force and presence in Gaza and a show of force they certainly did show," she said.
"[The rally] has surpassed all expectations [of the number of people attending]."
Hamas had set up checkpoints on the main north-south road in Gaza to check vehicles going to the event, residents said.
Their security forces also deployed across Gaza City and fired in the air on Monday morning at one intersection after youths threw stones at them, witnesses said.
Odeh said that she had spoken to people who had walked to the rally from Beit Hanoun, which lies at the northernmost tip of the Gaza Strip.
"Four months after the Hamas takeover of Gaza, the occasion of the anniversary of Yasser Arafat's passing has basically brought all Fatah members together to show that they are here," she said.
She said the rally had galvanised Fatah supporters after previous attempts to hold public demonstrations in the Gaza Strip were put down by Hamas's police force.
Fatah has had only a marginal presence in Gaza since Hamas forces violently took control of the region in June and took over key institutions.
The Wafa news agecny, which is run by the office of Mahmoud Abbas, the leader of Fatah and the Palestinian president, said Hamas had confiscated pictures of Arafat and headdresses symbolising the late leader.
The items were seized "at a time when the Gaza Strip is being deprived of basic goods and medicine", a Hamas security offical was quoted as saying by a pro-Hamas website.
Israel recently imposed caps on fuel imports to the Gaza Strip, leading to transport restrictions and reductions in electricity output to the Palestinian territory.
Source: Al Jazeera and agencies