Standing alongside the US president in the Roosevelt Room to address reporters, Abbas, speaking in Arabic, said he believed the parties are capable under Trump's "courageous stewardship and great negotiating ability" to bring about "a historic peace treaty".
Trump further said that that the two sides must continue to build their partnership to counter and defeat terrorism.
However, there have been no serious negotiations since gaps widened with the 2009 election of Benjamin Netanyahu as Israel's prime minister.
Mr Abbas, 82, who was among the negotiators on hand for the historic signing of the Oslo Accords on the White House lawn in 1993, indicated that he did feel optimistic that almost a quarter-century later, Mr Trump might be the president who finally builds on that initial agreement to forge a final resolution to the conflict.
US President Donald Trump has claimed that despite a vitriolic, religiously motivated war raging between Israel and Palestine spanning almost 70 years, officials of the two countries "work together beautifully".
Abbas pointed toward a likely partnership while crediting Trump's "great negotiating ability".
"President Abbas... How can you speak of peace with Israel and at the same time fund murderers who spill the blood of innocent Israelis at every turn?"
According to a Palestinian news agency the Palestinian delegation saw great importance in the White House meeting, both in terms of stabilizing Abbas's status as president and as the representative of the Palestinian people vis-à-vis the worldwide community.
Previously, tweets from President Trump's personal account have been sent out to mark different visits from various world leaders.
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Coach Scott Brooks called it the Wizards' best stretch this season, crediting defense for making it all happen. The physical series continues as the Wizards keep the flame alive in the series with a win-loss record of 2-1.
Abbas reasserted the goal of a Palestinian state, saying it must have East Jerusalem as its capital with the borders that existed before the 1967 Middle East war.
Trump said, flaunting his deal-making prowess during a first meeting with the veteran Palestinian leader that marked the start of a quixotic presidential effort.
"We made it clear that we support the efforts of President Trump ... and I hope that we will reach a solution", Abbas said. "We will get it done", Trump confidently told Abbas.
Welcoming Trump's role as a mediator in peace negotiations, the Palestinian leader said he looks forward to work with him in order to revive the stalled Israel- Palestine peace process. It isn't only "Israeli officials" who would feel that way, but any sentient being.
Abbas also does not appear to have secured US backing for pressure on Israel to end settlement construction in the lands the Palestinians claim for a future state - a major goal for the embattled Palestinian leader.
The hunger strike of more than 1500 Palestinian prisoners in Israeli jails was also highlighted on the front page of the dailies, particularly the rally held in Ramallah in support of the prisoners.
Earlier this week, his West Bank-based government announced it will stop paying for electricity Israel sends to power-starved Gaza.
Donald Trump hosted Mahmud Abbas in the White House for the conversation about a Middle East peace agreement.
Trump's promise to move the U.S. Embassy to Jerusalem from Tel Aviv, strongly opposed by Palestinians, has been shifted to the back burner, and he has asked Netanyahu to put unspecified limits on settlement activity.