The violence left three dead and more than 15 wounded in areas surrounding the presidential compound, bringing spokesmen for both parties to plead to their members for a cease-fire.
Gunmen in both parties agreed to a cease-fire, although questions remain whether the arrangement will hold.
On Saturday, Abbas, the leader to the Fatah Party, announced on live television that Hamas' inability to compromise and curb the recent fighting has reached a breaking point, and ordered early presidential and parliamentary elections. Elected in January 2006, the current Hamas-led parliament is supposed to remain in office through 2010.
After Abbas' announcement, the Hamas government released a statement rejecting Abbas' demand, calling it a "coup against Palestinian legitimacy." The group also said it would boycott the elections.
Palestinian officials also are debating whether or not Abbas even has the authority to dissolve the parliament. The Central Election Commission has been given the task of finding legal support for the early elections, according to the BBC.
Since the Hamas-elected takeover of the Palestinian Authority, the international community has cut much of its financial aid, sending the Palestinian people into deeper economic crisis and international isolation. In his speech, Abbas said, "We want to examine the will of the people. Do they still trust those they have chosen?"
In recent weeks, the violence in Gaza has become increasingly political in nature. Hamas Prime Minister Ismail Haniyeh avoided an assassination attempt while returning from Egypt; government officials and Hamas loyalists accused Abbas of being involved in the attack. On Sunday, the body of a top Fatah security officer was found in Northern Gaza.
Abbas' announcement also came on the eve of a diplomatic visit from British Prime Minister Tony Blair. At a joint press conference, Abbas insisted the elections would still take place, in spite of the violence.
"Your people are suffering," Blair said. "We don't want anything to stand in the way of helping the Palestinian people."
During the press conference, Abbas also spoke of the importance of brokering a peace agreement with Israel, saying of Israeli Prime Minister Ehud Olmert, "we need each other."
On Monday, Olmert announced that Israel would be ready to release funds for humanitarian needs in the Palestinian territories and that he would like to meet with Abbas "very soon."